# Correlated subgrain and particle analysis of a recovered Al-Mn alloy by
directly combining EBSD and backscatter electron imaging

Håkon W. Ånes<EMAIL_ADDRESS>Antonius T. J. van Helvoort  Knut Marthinsen
Department of Materials Science and Engineering, Norwegian University of
Science and Technology, 7491, Trondheim, Norway  Department of Physics,
Norwegian University of Science and Technology, 7491, Trondheim, Norway

###### Abstract

Correlated analysis of (sub)grains and particles in alloys is important to
understand transformation processes and control material properties. A
multimodal data fusion workflow directly combining subgrain data from electron
backscatter diffraction (EBSD) and particle data from backscatter electron
(BSE) images in the scanning electron microscope is presented. The BSE images
provide detection of particles smaller than the applied step size of EBSD down
to $0.03\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in diameter. The workflow is
demonstrated on a cold-rolled and recovered Al-Mn alloy, where constituent
particles formed during casting and dispersoids formed during subsequent
heating affect recovery and recrystallization upon annealing. The multimodal
dataset enables statistical analysis including subgrains surrounding
constituent particles and dispersoids’ location with respect to subgrain
boundaries. Among the subgrains of recrystallization texture,
Cube{001}$\left<100\right>$ subgrains experience an increased Smith-Zener drag
from dispersoids on their boundaries compared to CubeND{001}$\left<310\right>$
and P{011}$\left<\bar{5}\bar{6}6\right>$ subgrains, with the latter
experiencing the lowest drag. Subgrains at constituent particles are observed
to have a growth advantage due to a lower dislocation density and higher
boundary misorientation angle. The dispersoid size per subgrain boundary
length increases as a function of misorientation angle. The workflow should be
applicable to other alloy systems where there is a need for analysis
correlating grains and grain boundaries with secondary phases smaller than the
applied EBSD step size but resolvable by BSE imaging.

###### keywords:

Electron backscatter diffraction, texture analysis, particle analysis, image
registration, data fusion

††journal: Materials Characterization

##  1 Introduction

Electron backscatter diffraction (EBSD) is a technique to characterize
crystallographic features in the scanning electron microscope (SEM) [1]. It is
routinely used to characterize (sub)grain structures and textures in metals
[2], which strongly affect their mechanical properties. Secondary phases in
metals, like larger constituent particles or closely spaced fine dispersoid
particles, also strongly affect the mechanical properties of metals [3]. These
phases are often characterized by backscatter electron (BSE) imaging in the
SEM, utilizing the images’ mean atomic number contrast [4] which enable their
separation from the matrix phase. However, the analysis of grains and
secondary phases is typically performed on separate datasets which are not
directly spatially correlated. Without this correlation, the study of the
effect of individual particles on individual grains is difficult to automate
and upscale to become statistically significant. Furthermore, BSE imaging has
a higher spatial resolution than EBSD in general, of a few nm compared to
20-50 nm for EBSD, and can provide information on phases smaller than those
resolvable by EBSD.

Generally, in alloys containing a large number of secondary phase particles,
large constituent particles ($\geq$ $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$)
promote recrystallization by particle stimulated nucleation (PSN) [5], while
closely spaced fine dispersoids (sub-micron) inhibit the process by pinning
(sub)grain boundaries via Smith-Zener pinning [6, 7]. Dispersoid particles
precipitated concurrently with recrystallization in cold-rolled Al-Mn alloys
annealed either at a low temperature or non-isothermally have a pronounced
effect on their recrystallized grain structure and texture [8, 9, 10].
Observations include sluggish recrystallization, a coarse grain structure
elongated along the rolling direction, and an unusually sharp
P{011}$\left<\bar{5}\bar{6}6\right>$ texture component and a less sharp
CubeND{001}$\left<310\right>$ component, but still stronger compared to the
Cube{001}$\left<100\right>$ component. These observations are explained by a
growth advantage of P subgrains during recovery and recrystallization, assumed
to come about by their boundaries experiencing a reduced Smith-Zener drag than
those of CubeND and Cube subgrains [8, 10]. Direct statistically significant
evidence of this reduced drag has not yet been reported.

The goal of this work is to establish a data fusion workflow to directly
spatially correlate secondary phases, smaller than the applied EBSD step size,
and orientation mapping data of (sub)grains and (sub)grain boundaries. The
workflow is demonstrated on a 95% cold-rolled and non-isothermally annealed
Al-Mn alloy to a recovered state just before the onset of recrystallization.
The resulting multimodal dataset enables correlated analysis of dispersoid and
constituent particles and subgrains, and the results are presented and
discussed in the context of the effect of these particles on the three
recrystallization texture components P, CubeND, and Cube.

##  2 Methods

###  2.1 Material and sample preparation

The Al-Mn alloy, supplied by Hydro Aluminium, is direct-chill cast and has a
composition of Al–0.53Fe–0.39Mn–0.152Si wt.%. It is homogenized and cold-
rolled to 95% reduction. Huang et al. [10] studied the exact same alloy
applying the same homogenization and deformation, and found that it was fully
recrystallized after non-isothermal annealing from room temperature to
$350\text{\,}\mathrm{\SIUnitSymbolCelsius}$ at a rate of
$50\text{\,}\mathrm{\SIUnitSymbolCelsius}\text{\,}{\mathrm{h}}^{-1}$.
Preliminary analysis shows the same results for the current study. The
material has in this state a coarse and elongated grain structure and a
dominating P texture. In this work, the material is annealed to
$300\text{\,}\mathrm{\SIUnitSymbolCelsius}$ in an air furnace and water
quenched, again following the same annealing procedure as Huang et al.. The
material has in this state recovered and is on the onset of recrystallization.

The sample surface must be sufficiently free of deformation for EBSD analysis
and provide good phase contrast in BSE images to allow particle detection. The
sample surface inspected is the plane spanned by the rolling direction (RD)
and normal direction (ND). The sample embedded in epoxy is metallographically
polished using diamond paste down to $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$
grain size, followed by vibration polishing in a Buehler VibroMet 2 using
colloidal silica of $0.05\text{\,}\mathrm{\SIUnitSymbolMicro m}$ grain size.
The final deformed surface layer is removed with ion polishing in a Hitachi
IM-3000 using Ar+ ions accelerated to $3\text{\,}\mathrm{keV}$ with the sample
tilted $60\text{\,}\mathrm{\SIUnitSymbolDegree}$ from the horizontal and
rotating. The sample is plasma cleaned with a Fischione 1020 prior to
insertion in the SEM.

###  2.2 Acquisition of EBSD data and BSE images

Three EBSD datasets are acquired with a NORDIF UF-1100 detector on a Zeiss
Ultra 55 FEG SEM operated at an acceleration voltage of
$17\text{\,}\mathrm{kV}$. The acceleration voltage is a compromise between the
spatial resolution, which in general decreases with increasing voltage, and
the BSE yield on the detector, which increases with increasing voltage. The
sample is tilted $70\text{\,}\mathrm{\SIUnitSymbolDegree}$ from the horizontal
towards the detector, and the working distance is about
$24\text{\,}\mathrm{mm}$ for all datasets. All datasets are acquired from
nominally square regions of interest (ROIs) in the middle of the rolled slab
with a nominal step size $\Delta_{\mathrm{EBSD}}$ =
$0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The nominal area analyzed is
$0.021\text{\,}{\mathrm{mm}}^{2}$. The detector is binned by a factor of 6 to
a pattern resolution of (96 $\times$ 96) pixels of 8-bit depth. To determine
the average projection center (PC) for each dataset, five calibration patterns
of (240 $\times$ 240) pixels are collected from each corner and the center of
the ROI prior to acquisition. The acquisition speed is 70 or 75 patterns per
second. The EBSD acquisition parameters are chosen to both provide a
sufficient signal-to-noise ratio on the detector and a reasonable acquisition
time. Pattern analysis is detailed in Sec. 2.4.

BSE images covering each ROI are acquired after each EBSD acquisition. The
sample is reversed to a horizontal position, the working distance is reduced
to $5.4\text{\,}\mathrm{mm}$, and the acceleration voltage is reduced to
$5\text{\,}\mathrm{kV}$. Four BSE images with sufficient overlap between the
images are acquired. The acquisition time of each image is
$3\text{\,}\mathrm{min}$. The images are stitched into one image, henceforth
called the ‘BSE image’, using the ImageJ plugin BigStitcher [11]. The BSE
image appears free of stitching errors. Four images are acquired instead of
just one in order to use a smaller step size
$\Delta_{\mathrm{BSE}}\sim$$0.026\text{\,}\mathrm{\SIUnitSymbolMicro m}$. This
is important as the Al-Mn alloy contains dispersoids smaller than
$\Delta_{\mathrm{EBSD}}$. A smaller step size could be used to detect finer
dispersoids, but this would require more BSE images to be overlapped to cover
the ROI, increasing the total BSE image acquisition time and the risk of
stitching errors.

###  2.3 Multimodal data fusion workflow

This workflow has two goals: (1) correct distortions in the EBSD map and (2)
determine whether a point in the corrected EBSD map contains a particle. The
former is accomplished by image registration, while the latter is accomplished
by particle detection and data fusion accounting for the different step sizes
in the BSE image and EBSD map. The workflow is demonstrated on one of the
three datasets analyzed. Figures for the two remaining datasets are included
in the supplementary material.

####  2.3.1 Image registration

The goal of image registration [12] is to obtain a transformation function
which can interpolate between coordinates in a ‘reference’ image and a
‘sensed’ image, here the BSE image and EBSD map, respectively. Ideally,
information from the BSE image could be inserted directly into the EBSD map as
long as the difference in spatial resolution is handled by binning. However,
the EBSD map is generally distorted [13] compared to the BSE image, mostly due
to the high sample tilt and longer acquisition time. The thin plate spline
(TPS) transformation function [14, 12] previously used by Zhang et al. [15] to
correct distortions in EBSD maps is used for image registration in this work.

The general starting point of image registration is the reference and sensed
images of the same ROI showing features recognizable in both images [12]. The
images used here are the BSE image, which shows mostly mean atomic number
contrast, and an EBSD intensity map obtained by summing the intensities in
each raw EBSD pattern. A part of the full BSE image with the ROI and the EBSD
intensity map, from the first of the three datasets analyzed, are given in
Fig. 1 (a) and (b), respectively. The contrast in the EBSD intensity map is
stretched by clipping 1% of the lowest and highest intensities. Also presented
are the sample and detector positions relative to the electron beam during BSE
image and EBSD data acquisition.

Figure 1:  Control points used in image registration. 82 manually selected
control points in the same ROI in (a) the BSE image, (b) the EBSD intensity
map, and (c) their relative positions after scaling and translating the EBSD
control point coordinates. Positions of the sample and detectors relative to
the electron beam during BSE image and EBSD map acquisition are shown (not to
scale) below (a) and (b), respectively.

The TPS transformation function requires corresponding control points (CPs) in
the BSE image and EBSD map. Larger particles quite evenly distributed on the
sample surface are recognizable in both images in Fig. 1 (a, b), and so 82 CPs
are manually selected mostly from particle locations, plotted in (a, b). Not
all of the recognizable particles are selected as CPs; the selection of CPs is
discussed in Sec. 4.1. Relative positions of both sets of CPs are plotted in
Fig. 1 (c), where the CP coordinates in the BSE image are first downscaled to
the same scale as the EBSD map by dividing by
$\Delta_{\mathrm{EBSD}}/\Delta_{\mathrm{BSE}}$, and then translated by
subtracting the average shift between the CP coordinates in the BSE image and
the EBSD map. If the coordinates of $n$ CPs are denoted $(x_{i},y_{i})$ in the
BSE image and $(X_{i},Y_{i})$ in the EBSD map, the TPS transformation function
interpolating between the two sets of CPs is

$(X,Y)=a_{1}+a_{x}x+a_{y}y+\sum_{i=1}^{n}w_{i}U(\left|(x_{i},y_{i})-(x,y)\right|),$
(1)

where the kernel function is defined as $U(R)=R^{2}\log R^{2}$. The function
represents a plate of infinite extent deforming under point loads at the CPs
[12]. The first three terms, an affine transformation, define the plate best
matching all the CPs, while the last term, a weighted sum of radially
symmetric basis functions, represent bending forces provided by the $n$ CPs.
The unknown coefficients $a_{1}$, $a_{x}$, $a_{y}$, and $w_{i}$ are determined
based on the coordinates of the two sets of CPs, and the procedure is detailed
by Zhang et al. [15]. We determine them using the TPS transformation function
implemented in the Python package morphops v0.1.13 [16], which is based on the
work by Bookstein [14]. The inverse mapping from the downscaled BSE image
coordinates $(x_{i},y_{i})$ to the EBSD map coordinates $(X_{i},Y_{i})$ is
then found by inserting $(x_{i},y_{i})$ into Eq. (1) together with the
determined coefficients. This mapping allows points in the EBSD map to be re-
mapped to the scaled BSE image coordinates by rounding to the closest map
coordinate, which requires no interpolation of information in the original
EBSD map points, like orientations.

The EBSD intensity map before and after correction is shown in Fig. 2 (a) and
(b), respectively. A rectangular grid with a nominal spacing of
$5.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is attached to the EBSD intensity
map before correction in order to highlight the distortions, and the grids
before and after correction are presented in (c). The nominal area analyzed is
$8446\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$, but the actual area
scanned is $8123\text{\,}{\mathrm{\SIUnitSymbolMicro m}}^{2}$, which is a 4%
reduction. The actual area scanned was 7% greater compared to the nominal area
for the other two datasets, demonstrating that the distortions are not
constant between scans.

Figure 2:  EBSD intensity map before and after correction using the TPS
transformation function and the CPs in Fig. 1. The EBSD intensity map with a
grid attached (a) before correction and (b) after correction, and (c) the grid
before and after correction.

####  2.3.2 Detection of particles and data fusion

Knowing the coordinates of each EBSD map point in the BSE image, it is
possible to insert information from the latter in the EBSD map. To utilize the
higher fidelity of the BSE image, particles are detected in the BSE image in
its original resolution, also outside the EBSD map ROI to improve the
statistical analysis of particle size and volume fraction. This produces a
binary particle map, and for the ROI in this particle map to fit into the EBSD
map, it must be binned. If the binned particle map is to remain binary, the
binning factor must be an integer. The different spatial resolutions
$\Delta_{\mathrm{EBSD}}$ and $\Delta_{\mathrm{BSE}}$ do not allow this, so
prior to particle detection, the BSE image is upscaled to allow integer
binning. The integer binning factor is rounded to the nearest upper integer
$x_{\mathrm{bin}}=\lceil\Delta_{\mathrm{EBSD}}/\Delta_{\mathrm{BSE}}\rceil$,
and the upscaling factor is
$x_{\mathrm{bin}}\cdot(\Delta_{\mathrm{BSE}}/\Delta_{\mathrm{EBSD}})$.

Particles are detected in the upscaled BSE image using region-based
segmentation as implemented in the Python package scikit-image v0.18 [17].
Particles are detected in the BSE images from all three datasets following six
steps with all parameters fixed: (1) Remove long-range intensity variations by
subtracting a Gaussian blurred version of the image, and normalize the image
intensities; (2) Generate an elevation map using the amplitude of the
intensity gradient obtained by correlating the BSE image with a Sobel filter;
(3) Determine markers of points that can unambiguously be labelled as particle
or background (subgrains) in the image by applying a threshold found from the
intensity distribution; (4) Segment particles with the watershed transform
using the elevation map and markers; (5) Remove holes within segmented
particles, as they are assumed to contain no holes; (6) Finally, remove
incorrectly detected particles by assuming that correctly detected particles
have (i) sufficiently high roundness, defined as the ratio between the
perimeter of the convex hull and the actualt perimeter, (ii) sufficiently high
solidity, defined as the ratio between the area of the convex hull and the
actual area, and (iii) sufficiently high mean BSE image intensity compared to
the background. Based on these assumptions, particles which satisfy any of the
following three criteria are kept: (i) roundness greater than the 25
percentile, (ii) solidity greater than the 25 percentile, and (iii) mean BSE
image intensity greater than the 55 percentile. By identifying some correctly
and incorrectly detected particles, the percentile thresholds are found
manually in a search so that as many of the former and as few of the latter
are kept.

To get a qualitative assessment of how good the image registration and
particle detection is, particles are detected in the EBSD intensity map in
Fig. 1 (a) following the steps (1-5) above. The 4972 particles detected in the
BSE image and the 217 particles detected in the EBSD intensity map are shown
in Fig. 3 (a, b), respectively, while the particles are given in (c), with
overlapping particles colored black. Many of the larger particles detected in
both images overlap, accounting for 36% of colored pixels in (c). This
indicates a reasonably successful image registration and that particles are
correctly detected, assuming a similar mean atomic number contrast in both
images. There are thousands of small particles and some larger ones only
detected in the BSE image, visible as pink in (c), accounting for 55% of
colored pixels. The 9% of colored pixels only detected in the EBSD intensity
map, visible as green in (c), are mostly found in pixels surrounding particles
detected in both the BSE image and the EBSD intensity map. This can be
explained by the fact that the detected signal on the EBSD detector originates
from a larger interaction volume due to the higher acceleration voltage and
the $70\text{\,}\mathrm{\SIUnitSymbolDegree}$ sample tilt during EBSD data
acquisition.

Figure 3:  Particles detected with region-based segmentation in the ROI
defined by the corrected EBSD map. Particles detected in (a) the BSE image and
(b) the corrected EBSD intensity map, and (c) the two particle sets
overlapped, with overlapping regions shown in black.

Before inserting the upscaled binary particle map into the EBSD map, the
particle map is binned by the integer binning factor $x_{\mathrm{bin}}$,
meaning blocks of ($x_{\mathrm{bin}}\times x_{\mathrm{bin}}$) pixels are
binned into one pixel. With $x_{\mathrm{bin}}=4$, particles constituting one
pixel and 16 pixels in the unbinned particle map might be given the same
‘size’ of one pixel in the binned particle map, which will distort the
analysis of particle size from the binned particle map. To account for this, a
binned map containing the number of particle pixels within each (4 $\times$ 4)
block is included alongside the binned binary particle map in the EBSD data
and used when calculating the particle size. Another caveat is that if parts
of two neighboring particles are binned into the same point, the number of
particles in that point is two. This fact is neglected when binning in this
work, as it is only the case for fewer than 80 of more than two million points
in all three datasets. In summary, the final result of the multimodal data
fusion workflow is the corrected EBSD map containing the binned binary
particle map and the map of the number of particle pixels in each map point.

###  2.4 Analysis of EBSD patterns and orientations

The crystal orientations from each EBSD pattern in the original EBSD maps are
determined by searching for the best match in a discrete dictionary of
simulated patterns by dictionary indexing (DI) [18], as implemented in the
Python package kikuchipy v0.5 [19, 20]. The basis for DI is a dynamically
simulated master pattern, simulated for Al (space group $Fm\bar{3}m$) with the
EMsoft v4.3 [21] Fortran suite of programs. Simulated patterns in the
dictionary are projected from the master pattern onto the EBSD detector using
an average PC per dataset, found from the Al calibration patterns using Hough
indexing in the Python package PyEBSDIndex [22]. The orientations of the
simulated patterns are sampled using cubochoric sampling [23] as implemented
in the Python package orix v0.7 [24, 25]. The orientations are uniformly
distributed in the fundamental zone with an average misorientation angle of
$1.4\text{\,}\mathrm{\SIUnitSymbolDegree}$, resulting in a dictionary of about
300 000 simulated patterns. The signal-to-noise ratio of the raw experimental
patterns is increased prior to indexing by subtracting a static background,
followed by subtraction of a dynamic (per pattern) background, and finally
averaging each pattern with its eight nearest neighbors using a Gaussian
kernel with a standard deviation of 1, as implemented in kikuchipy v0.5. Every
experimental pattern is compared to the dictionary via the normalized cross
correlation (NCC) coefficient $r$ [26], and the orientation of the simulated
pattern with the highest $r$ is chosen as the solution per pattern. These
discrete orientations are refined by letting them vary while keeping the PC
fixed, and optimizing $r$. The Nelder-Mead optimization algorithm as
implemented in the Python package SciPy v1.7 [27] is used.

Orientation analysis is done with the MATLAB toolbox MTEX v5.8 [28] on the
corrected EBSD map containing the particle map. No averaging or clean-up of
orientations is applied. The particle map is used to create a ‘dual phase’
dataset to distinguish between points with Al and points with particles.
Subgrains are reconstructed with a misorientation angle threshold of
$1\text{\,}\mathrm{\SIUnitSymbolDegree}$, with subgrains smaller than five
points assigned to neighboring subgrains. Grain boundaries are smoothed in
five iterations, resulting in a reduced total boundary length and allowing
boundary segments to have coordinates independent of the rectangular grid of
the corrected EBSD map. This results in two ‘grain’ populations of Al
subgrains and particles. Typical rolling and recrystallization texture
components in Al [29] considered in this work are listed in Table 1. The
subgrains are classified into these texture components based on the lowest
misorientation angle between the subgrain mean orientation and the components’
ideal orientations within a threshold of
$15\text{\,}\mathrm{\SIUnitSymbolDegree}$. Subgrains outside the threshold are
considered random and termed ‘other’ in the remainder of the text. An
orthorhombic sample symmetry valid for cold-rolled samples is assumed when
assigning texture components. Grain sizes $D$ and particle sizes $d$ are given
as equivalent circular diameters $0.816\cdot 2\cdot\sqrt{A/\pi}$, where $A$ is
the grain or particle area and the prefactor is a result of stereological
considerations [29].

Table 1: Typical texture components in cold-rolled Al [29] represented by Miller indices {hkl}$\left<uvw\right>$. In general, Br, Cu and S are deformation textures and the remaining are recrystallization textures. Name | {hkl}$\left<uvw\right>$
---|---
Br | {011}$\left<2\bar{1}1\right>$
Cu | {112}$\left<11\bar{1}\right>$
S | {123}$\left<634\right>$
Cube | {001}$\left<100\right>$
CubeND | {001}$\left<310\right>$
Goss | {011}$\left<100\right>$
P | {011}$\left<\bar{5}\bar{6}6\right>$

##  3 Results

###  3.1 Multimodal dataset of subgrains and particles

The result of the multimodal data fusion workflow, the Al-Mn microstructure
represented in terms of Al subgrains and particles, is shown in Fig. 4 for the
first of the three datasets. The Al orientation map colored according to the
direct lattice vector parallel to RD is presented in (a, d). Subgrains colored
by texture component are shown in (b, e), with low–angle grain boundaries
(LAGB) with boundary misorientation angles $\omega\in$
[$1\text{\,}\mathrm{\SIUnitSymbolDegree}$,
$15\text{\,}\mathrm{\SIUnitSymbolDegree}$) and high–angle grain boundaries
(HAGB) with $\omega\geq$ $15\text{\,}\mathrm{\SIUnitSymbolDegree}$ colored
gray and black, respectively. Particles are colored black in (a, b, d, e).
Bands of constituent particles along RD are indicated with arrows in (a, b),
where most of the subgrains of random orientations or recrystallization
textures are located. A magnified view of the microstructure is shown in the
BSE image and orientation and grain maps in (c-e), respectively, illustrating
the successful detection and localization of particles in the subgrain
structure. The microstructure is recovered, and no obvious recrystallization
nuclei are observed in the three datasets studied.

Figure 4:  Multimodal dataset of the Al microstructure with subgrains and
particles. (a) Al orientation map colored according to the direct lattice
vector parallel to RD (indicated bottom left), and (b) subgrains colored by
texture component according to the legend below. Subgrains not assigned a
texture component are white. The highlighted region is magnified in (c) the
BSE image and the (d) orientation and (e) grain maps. Particles are colored
black in (a, b, d, e), while LAGB and HAGB are respectively colored gray and
black in (b, e). Arrows in (a, b) highlight bands of constituent particles.

The combined subgrain and particle statistics from all three datasets are
given in Fig. 5. The volume fraction $F_{\mathrm{V}}$ of subgrains of rolling
texture is 72% as shown in (a), typical of an Al alloy deformed by cold-
rolling but not yet recrystallized. 15282 subgrains make up this volume
fraction given in (b), with a mean area weighted grain size of
$D_{\mathrm{A,\>roll.}}$ = $1.59\pm 0.02\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ as shown in (c). More interesting when studying recrystallization are the
782 subgrains of recrystallization texture constituting a volume fraction of
3%, with a mean grain size of $D_{\mathrm{A,\>rex.}}$ = $1.15\pm
0.03\text{\,}\mathrm{\SIUnitSymbolMicro m}$. The remaining grain volume
fraction are 6777 subgrains of other orientations with an average grain size
of $D_{\mathrm{A,\>other}}$ = $1.16\pm 0.01\text{\,}\mathrm{\SIUnitSymbolMicro
m}$. The uncertainties are the 95% confidence interval of the mean, and this
is used throughout the text when uncertainties are given.

Figure 5:  Subgrain and particle statistics from all three datasets combined.
(a) Volume fraction $F_{V}$, (b) number of subgrains, and (c) area weighted
grain size $D_{A}$, per texture component. (d) Area weighted histogram and
cumulative probability distribution of particle sizes $d$, with size regions
classified as dispersoids and constituent particles indicated.

A histogram of the area weighted particle size $d$ of about 90000 particles
detected in the BSE images from all three datasets, including outside the EBSD
ROIs, is shown in Fig. 5 (d). About 11000 of the particles are inside the EBSD
ROIs. The cumulative probability distribution of the particle sizes $d$
plotted in (d) shows that about 91% of particles are smaller than
$0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in diameter, which constitutes 1
point in the EBSD maps and 17 pixels in the BSE images. Dispersoids are here
defined as those smaller than $0.24\text{\,}\mathrm{\SIUnitSymbolMicro m}$,
seen in Fig. 5 (d). This threshold is chosen based on the somewhat bimodal
distribution of $d$. The critical diameter for particle stimulated nucleation
(PSN) after rolling to 95% is $\sim$$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$
[5], and $0.8\text{\,}\mathrm{\SIUnitSymbolMicro m}$ is used here as the lower
threshold for particles defined as constituent particles, which are considered
potential nucleation sites for recrystallized grains. Particle parameters for
all detected particles, constituent particles, and dispersoids are given in
Table 2, assuming particles to be spherical and randomly distributed.

Table 2:  Particle parameters: volume fraction $f_{\mathrm{V}}$, area weighted average particle size $d_{\mathrm{A}}$, and interparticle spacing $N_{\mathrm{s}}^{-0.5}$.  Population |  $f_{\mathrm{V}}$ [%] |  $d_{\mathrm{A}}$ [$\mathrm{\SIUnitSymbolMicro m}$] |  $N_{\mathrm{s}}^{-0.5}$ [$\mathrm{\SIUnitSymbolMicro m}$]
---|---|---|---
All | 1.84 | $0.802\pm 0.001\text{\,}$ | 0.306
Constituent | 0.73 | $1.400\pm 0.079\text{\,}$ | 10.852
Dispersoids | 0.31 | $0.089\pm 0.001\text{\,}$ | 0.723

###  3.2 Dispersoids at subgrain boundaries

By computing the Euclidean distance from a particle to every subgrain boundary
segment within a square of (2 $\times$ 2) $\mathrm{\SIUnitSymbolMicro m}$
centered on the particle, the minimum distance from each particle to a segment
and the number of particles within a certain distance from a segment can be
determined. The cumulative probability of dispersoids’ distance to the closest
boundary is given in Fig. 6 (a): 60% of dispersoids are within
$0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of a boundary, the same distance
as $\Delta_{\mathrm{EBSD}}$, and these dispersoids are considered to be at
boundaries. The remaining dispersoids are within subgrains. The farthest a
dispersoid is from a boundary is about $1.3\text{\,}\mathrm{\SIUnitSymbolMicro
m}$, which is reasonable considering that the average subgrain size
$D_{\mathrm{A}}$ is of the same magnitude. Two parameters are considered based
on the dispersoids at boundaries: The number of dispersoids per boundary
length, given as dispersoids ${\mathrm{\SIUnitSymbolMicro m}}^{-1}$, and the
dispersoid size $d$ per boundary length, which is dimensionless.

Classifying boundary segments by their misorientation angle $\omega$, the
number of dispersoids per boundary length per $\omega$ is plotted in Fig. 6
(b). There is on average 0.26 dispersoids ${\mathrm{\SIUnitSymbolMicro
m}}^{-1}$, i.e. about 1 dispersoid every $4\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ of boundary, and the amount stays constant with increasing $\omega$. No
correlation is observed when inspecting the misorientation axis either. The
dispersoid size $d$ per boundary length as a function of $\omega$ is plotted
in (d). There is on average about one dispersoid of size
$0.022\text{\,}\mathrm{\SIUnitSymbolMicro m}$ per
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of boundary, and here the amount
increases slightly with increasing $\omega$.

Classifying boundary segments by the texture component on either side of the
boundary, the number of dispersoids per boundary length per texture component
is plotted in Fig. 6 (c): There is no significant difference for the various
subgrain boundaries, apart from slightly more dispersoids on CubeND and Goss
subgrain boundaries. Looking at the dispersoid size $d$ per boundary length as
a function of texture component in (e), P subgrains stand out among the
subgrains of recrystallization textures as having significantly larger
dispersoids on their boundaries.

When considering only HAGBs, an insignificant difference in the number of
dispersoids on HAGBs of subgrains at constituent particles and subgrains
elsewhere is observed: the former boundaries have on average
$0.239\pm 0.060\text{\,}\mathrm{d}$ispersoids ${\mathrm{\SIUnitSymbolMicro
m}}^{-1}$ while the latter boundaries have on average $0.248\pm
0.007\text{\,}\mathrm{d}$ispersoids ${\mathrm{\SIUnitSymbolMicro m}}^{-1}$.
The dispersoid size $d$ per boundary length is slightly lower but still
insignificant, with $0.022\pm 0.006\text{\,}\mathrm{f}$or HAGBs at constituent
particles and $0.026\pm 0.001\text{\,}\mathrm{f}$or HAGBs elsewhere.

P, CubeND and Cube subgrains all have orientation relationships to the
deformed matrix close to the special misorientation of
$38\text{\,}\mathrm{\SIUnitSymbolDegree}$$\left<111\right>$ [30], or $\Sigma
7$ in coincidence site lattice terminology, which represents a boundary with a
reduced energy and a higher migration rate than general HAGBs [29]. A
significant difference in the number of dispersoids per boundary length is
observed between boundaries of $\omega\leq$
$15\text{\,}\mathrm{\SIUnitSymbolDegree}$ to $\Sigma 7$ and other boundaries:
the former boundaries have on average $0.233\pm
0.017\text{\,}\mathrm{d}$ispersoids ${\mathrm{\SIUnitSymbolMicro m}}^{-1}$
while the latter boundaries have on average $0.263\pm 0.005\text{\,}\\\ $
dispersoids ${\mathrm{\SIUnitSymbolMicro m}}^{-1}$. Furthermore, the
dispersoid size $d$ per boundary length is higher for $\Sigma 7$ boundaries
than other boundaries, with $0.026\pm 0.002\text{\,}\mathrm{c}$ompared
$0.022\pm 0.001\text{\,}$ 13-18% of boundaries of subgrains of
recrystallization textures are of $\Sigma 7$-type, while less than 9% of
boundaries of subgrains of rolling textures and other orientations are of this
type.

Figure 6:  Dispersoids at subgrain boundaries. (a) Cumulative probability
distribution of dispersoids’ distance to the nearest subgrain boundary.
Dispersoids with distances below $0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$
are considered to be at boundaries. (b) Dispersoids per boundary length as a
function of boundary misorientation angle. (c) Dispersoids per boundary length
per texture component. (d) Dispersoid size $d$ per boundary length as a
function of boundary misorientation angle. (e) Dispersoid size $d$ per
boundary length per texture component. Linear fits are shown in (b, d) while
95% confidence intervals are shown in (b-e).

###  3.3 Subgrains at constituent particles

Analysis of subgrains at constituent particles is presented in Fig. 7. As
shown in (a), most of these subgrains are randomly oriented, which is
generally found for subgrains in deformation zones at constituent particles in
heavily deformed polycrystals [29]. This is also evident from Fig. 4 (b) by
observing that randomly oriented subgrains are mostly found in particle bands
[31]. Subgrains of recrystallization textures at constituent particles are
mostly CubeND, Goss, and P, with fewest Cube subgrains. Examples of subgrains
of recrystallization textures at constituent particles are seen in Fig. 4 (d,
e). Fig. 7 (a) shows that the sizes of subgrains per texture component at
constituent particles are in general smaller than subgrains found elsewhere in
the microstructure, in accordance with previous comparisons of size
distributions of subgrains close to and far away from constituent particles
[5]. The amount of plastic strain within a grain can be approximated by the
geometrically necessary dislocation density $\rho_{\mathrm{GND}}$, calculated
from orientation differences in neighboring map points using Pantleon’s
approach [32] implemented in MTEX. As was done by Pantleon, the
{111}$\left<110\right>$ slip system is assumed active with 12 edge
dislocations with line energies of 1 and 6 screw dislocations with line
energies of 1 - $\nu$, where $\nu$ = 0.347 is the Poisson ratio for Al.
Distributions of $\rho_{\mathrm{GND}}$ in subgrains at constituent particles
and elsewhere are presented in Fig. 7 (b). Distributions of the boundary
misorientation angle $\omega$ for the same two groups of subgrains are given
in Fig. 7 (c). Subgrains at constituent particles have lower dislocation
densities and higher $\omega$ compared to subgrains elsewhere. This indicates
that the former subgrains have recovered more than the latter subgrains, and
that they have favorable conditions for becoming recrystallization nuclei,
i.e. they have a growth advantage. Two other commonly used subgrain
orientation parameters, grain orientation spread and grain average
misorientation, neither shown here, give a similar picture to that of
$\rho_{\mathrm{GND}}$.

Figure 7:  Subgrains at constituent particles. (a) Number of subgrains at
constituent particles (left), and the ratio of mean area weighted size of
subgrains at constituent particles $D_{\mathrm{A,\>constituent}}$ over all
grain sizes $D_{\mathrm{A}}$ (right), per texture component. (b) Normalized
and area weighted histograms of $\rho_{\mathrm{GND}}$ in subgrains at
constituent particles and subgrains elsewhere, with the means. (c) Normalized
histograms of the boundary misorientation angle $\omega$ of subgrains at
constituent particles and subgrains elsewhere, with the means.

##  4 Discussion

###  4.1 Evaluation of the multimodal data fusion workflow

The presented multimodal data fusion workflow accomplishes two tasks: first,
it corrects distortions in the EBSD map [13], and secondly, it fuses data from
the particle map of higher fidelity with the corrected EBSD map. There are
numerous available solutions to the first task [15, 33, 34], with various
degrees of complexity. To accomplish the first task, the present workflow
requires a minimum of two images showing shared features, but which can have
totally different contrast and intensity, manual selection of a sufficient
number of CPs evenly distributed in the images, and a single transformation
function without tuning parameters to correct distortions in the EBSD map
based on the CPs. The most critical step in the workflow is the selection of
CPs.

Zhang et al. [15] studied the effect of the number of CPs used in the TPS
transformation function to correct an EBSD map, similar to the maps in this
work, with $\Delta_{\mathrm{EBSD}}$ = $0.2\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ and ROI area (120 $\times$ 120) ${\mathrm{\SIUnitSymbolMicro m}}^{2}$.
They found that correction of the EBSD map improved substantially when using
54 CPs compared to 36, and improved minimally when using 60 CPs compared to
54. The improvement was quantified in two ways: by comparing the outline of
the corrected EBSD maps to the same outline visible as a contaminated area in
a BSE image obtained with no sample tilt taken as the ground truth, and by
inspecting non-overlapping map points in overlapped Kikuchi band contrast maps
from the corrected EBSD maps obtained when using 36 and 54 CPs to the same map
when using 60 CPs, taken as the ground truth. The surface contamination
resulting from the EBSD acquisition is not visible in the BSE images in this
work. Since the end goal of image registration here is data fusion, maximizing
the similarity between the reference and sensed images leads to a more correct
multimodal dataset. A similar reasoning was used by Nguyen and Rowenhorst [35]
in image registration and data fusion of pores from BSE images with EBSD data
in an additive manufactured steel sample, when a control point-free
transformation function was found iteratively by optimizing the similarity of
binary maps of pores detected in the BSE images and EBSD maps. By assuming
that the BSE image and EBSD intensity map in Fig. 1 (a, b) have similar
contrast, and by rescaling the corrected EBSD map to the same resolution as
the BSE image and only comparing the ROI, a measure of their similarity is
obtained via the NCC coefficient $r$. The similarity between the corrected
EBSD intensity map in Fig. 2 (b) and the BSE image in Fig. 1 (a) when using
all the 82 CPs is $r_{82}$ = 0.3625. By using 77 CPs, chosen at random 30
times, a slightly worse $r_{77}=0.3622\pm 0.0003$ is obtained, and by
repeating this calculation but instead using 72 CPs, $r_{72}=0.3620\pm 0.0006$
is obtained. Using only 7 CPs results in $r_{7}=0.2400\pm 0.0159$. Inspection
of the relative error $100\cdot(r_{i}-r_{i-5})/r_{i}$ where $i=82,77,72,...,7$
shows that it increases minimally from 0.08 when using 77 CPs to 0.89 when
using 32 CPs, but increases substantially to 1.69 when using only 27 CPs, and
continues to increase substantially when using fewer CPs. This indicates that
about 30 CPs might be sufficient for registering the EBSD and BSE data, while
fewer than 30 CPs would give an incorrect registration and thus an incorrect
basis for correlated analysis. The absolute values for $r$ are different for
the two other datasets, but the trends are the same. As was found by Zhang et
al. [15], these results show that correction is improved by increasing the
number of CPs. Plots of $r$ and the relative error in $r$ as functions of the
number of control points used in image registration are included in the
supplementary material.

###  4.2 Effects of particles on recovery and recrystallization

Recrystallization textures are in general explained by nucleation and growth
advantages of grains of certain orientations [36]. The grain data alone shows
that Cube subgrains have a size advantage over CubeND and P subgrains since
they are respectively 43% and 33% larger. A possible explanation for this
observation is that Cube subgrains have experienced faster recovery and
started to grow earlier than CubeND and P subgrains [37, 38], however no
significant difference in the average $\rho_{\mathrm{GND}}$ between these
subgrain populations is observed. CubeND and P subgrains, on the other hand,
have a nucleation advantage, with respectively 214% and 326% more subgrains
than Cube subgrains. The combined subgrain and particle data shows that CubeND
subgrains have more dispersoids on their boundaries, while P boundaries are
populated by larger dispersoids than the other boundaries of subgrains of
recrystallization textures. The Smith-Zener drag from randomly distributed
dispersoids of average radius $d/2$ on near-planar grain boundaries of average
energy $\gamma$ is given by $P_{\mathrm{SZ}}=3\gamma f_{\mathrm{V}}/d$ [39,
29]. The interparticle spacing $N_{\mathrm{s}}^{-0.5}$ of dispersoids in Table
2 is about half of the average subgrain size $D_{\mathrm{A}}$, and there is a
strong subgrain boundary–dispersoid correlation since 60% of dispersoids are
found at boundaries. For these reasons, a modified expression for
$P_{\mathrm{SZ}}$ must be used [29]. Hutchinson and Duggan [40] argued that
when a large fraction of the particles lies at subgrain boundaries, the drag
force per unit area of subgrain boundary (sb) can be rewritten as
$P_{\mathrm{sb}}=3\gamma_{\mathrm{sb}}f_{\mathrm{V}}/(A_{\mathrm{sb}}d^{2})$,
where $A_{\mathrm{sb}}$ is the area of subgrain boundary per unit volume and
$\gamma_{\mathrm{sb}}$ is the average subgrain boundary energy. If the average
subgrain boundary edge length is $L$, we can assume $A_{\mathrm{sb}}=3/L$ [29]
to get $P_{\mathrm{sb}}=\gamma_{\mathrm{sb}}f_{\mathrm{V}}L/d^{2}$. By
expressing the dimensionless average dispersoid volume fraction
$f_{\mathrm{V}}$ as $f_{\mathrm{L}}d/2$ per texture component, where
$f_{\mathrm{L}}$ is the dispersoids per boundary length plotted in Fig. 6 (b,
c), and by approximating $L$ by the area weighted average subgrain size
$D_{\mathrm{A}}$, we arrive at the Smith-Zener drag at subgrain boundaries per
texture component

$P_{\mathrm{sb}}^{\prime}=\frac{\gamma_{\mathrm{sb}}f_{\mathrm{L}}D_{\mathrm{A}}}{2d}\propto\frac{f_{\mathrm{L}}D_{\mathrm{A}}}{d},$
(2)

where differences in $\gamma_{\mathrm{sb}}$ is neglected.
$P_{\mathrm{sb}}^{\prime}$ per texture component is plotted in Fig. 8 with the
confidence intervals of $P_{\mathrm{sb}}^{\prime}$ obtained from error
propagation of the confidence intervals of $f_{\mathrm{L}}$, $D_{\mathrm{A}}$,
and $d$. Among subgrains of recrystallization texture, Cube subgrains
experience the highest drag while P subgrains experience the lowest drag, with
CubeND subgrains in between. These results help to understand the dominating P
texture observed when concurrent precipitation strongly affects the
recrystallization texture and grain size, as is the case in the material
studied here [8, 9, 10]. It is worth noting that P subgrains have the largest
dispersoids on their boundaries, normalized by boundary length, as evident
from Fig. 6.

Figure 8:  Smith-Zener drag at subgrain boundaries per texture component as
calculated from Eq. (2).

##  5 Concluding remarks

A multimodal data fusion workflow enabling spatially correlated subgrain and
particle analysis of alloys in the SEM is presented. Image registration of BSE
images and EBSD maps is accomplished using the thin plate spline
transformation function with a sufficient number of control points evenly
distributed over the ROI. Registration and subsequent data fusion results in a
distortion corrected EBSD map containing the locations and sizes of particles,
many of them smaller than the applied EBSD step size, down to
$0.03\text{\,}\mathrm{\SIUnitSymbolMicro m}$ in diameter. The workflow is
demonstrated on data from a cold-rolled and recovered Al-Mn alloy just before
the onset of recrystallization, and results in the following findings:

  1. 1.

P and Cube subgrains experience respectively the lowest and highest Smith-
Zener drag from dispersoids on their boundaries among the subgrains of
recrystallization texture, which helps to explain the strong P texture in this
alloy when non-isothermally annealed or annealed at a low temperature,
observed by many.

  2. 2.

Subgrains at constituent particles have a growth advantage due to a lower
dislocation density, higher boundary misorientation angle $\omega$ and fewer
dispersoids on their boundaries.

  3. 3.

The dispersoid size per boundary length increases as a function of $\omega$,
while the number of dispersoids per boundary length is constant as a function
of $\omega$.

The workflow should be applicable to other alloy systems where quantitative
correlated analysis of minor secondary phases (of sizes down to and less than
$0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}$) and orientations of (sub)grains
is important to understand microstructure evolution.

## Data and code availability

The raw EBSD datasets and BSE images required to reproduce these findings are
available from Zenodo at https://doi.org/10.5281/zenodo.6470217 [dataset]
[41]. All softwares used for analysis of the data, apart from MTEX which
requires a MATLAB license, are freely available. Jupyter notebooks for all
processing steps and MTEX scripts can be obtained from
https://github.com/hakonanes/correlated-grains-particles-workflow.

## Acknowledgements

HWÅ acknowledges NTNU for financial support through the NTNU Aluminium Product
Innovation Center (NAPIC). The authors would like to thank Hydro Aluminium for
supplying the material.

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