# Habit plane determination from reconstructed parent phase orientation maps

Tuomo Nyyssönen  Azdiar A. Gazder  Ralf Hielscher  Frank Niessen  Swerim AB,
164 40 Kista, Sweden  Electron Microscopy Centre, University of Wollongong,
New South Wales 2500, Australia  Freiberg University of Mining and Technology,
09599 Freiberg, Germany  Technical University of Denmark, Department of Civil
and Mechanical Engineering, 2800 Kgs. Lyngby, Denmark

###### Abstract

This study details the development and validation of a new algorithm that
determines the dominant habit plane of a transformed child phase from
orientation maps of a single planar cross-section. The method describes the
habit plane in terms of its five-parameter grain boundary character and
couples it to the specific orientation relationship of the identified
orientation variant. The symmetry operations associated with the specific
orientation relationship of the variants are applied to transform habit plane
traces as determined in the specimen-fixed reference frame into the parent or
child reference frame, allowing for the fitting of the habit plane. Our
algorithm stands out by its robustness, computational efficiency, automation
and ability to operate on fully transformed microstructures. Four automated
methods for habit plane trace determination are proposed and compared.
Detailed sensitivity analysis reveals that the proposed algorithm is
exceptionally robust against poor accuracy in the measured traces and
distortions in the orientation map, but more sensitive to inaccuracies
propagated from parent grain reconstruction. Validation on a synthetic
microstructure with a known habit plane and returned consistent results when
applied to high and low carbon steels with different prior austenite grain
sizes and orientation map resolutions. The habit planes were not significantly
affected by the austenite grain sizes. The habit plane of the steel with 0.35
wt.% C was close to $(111)_{\gamma}$ whereas the habit plane of steel with
0.71 wt.% C was closer to $(575)_{\gamma}$, in close agreement with previous
work using two-surface stereological analysis and transmission electron
microscopy-based trace analysis.

###### keywords:

habit plane , phase transformation , parent grain reconstruction , orientation
relationship , electron backscattering diffraction (EBSD)

##  1 Introduction

The transformation of a parent phase into a child phase may occur by a
diffusive or shear-controlled transformation [1]. In crystalline materials,
phase transformations generally involve an orientation relationship that
minimizes the lattice mismatch at the interface. When the growth of the child
phase is constrained, the interface between parent and child phases may form
short coherent ledges that, when observed macroscopically, yield an irrational
crystal plane that is defined as the habit plane. The determination of the
habit plane has historically attracted attention in martensitic steels since
calculating and characterizing the orientation relationship and habit plane
were the only means of characterizing the rapidly occurring and
technologically important phase transformation [2]. Today, habit plane
determination in martensitic steel remains one of the most challenging tasks
in metallurgy due to the high crystal symmetry and the fully transformed,
complex hierarchical, and oftentimes fine-grained microstructures. Thus,
although the methods developed in this study are generally applicable to any
phase transformation system, the emphasis is on the particularly challenging
case of lath martensitic steel.

The morphology of martensite in steels can be changed from laths to plates by
increasing the carbon content, which goes hand-in-hand with a characteristic
change in the habit plane [3]. The shape deformation of the austenite-to-
martensite ($\gamma$-to-$\alpha^{\prime}$) transformation in steels leads to a
rotation of the macroscopic habit plane away from the rational
$(111)_{\gamma}||(110)_{\alpha^{\prime}}$ interface at the atomic level. In
1949 Greninger and Troiano [4] observed that the shape deformation of the
martensitic transformation resembles an invariant plane strain, i.e. a
homogeneous strain on the habit plane. However, they realized that this shear
cannot transform the parent austenitic lattice into the child martensitic one.
Alternatively, the well-known Bain strain [5] is an invariant plane strain
that transforms the austenitic lattice into the martensitic one, but distorts
the habit plane. Therefore, the transformation strain needed a combination of
two invariant plane strains, which in turn motivated the development of the
Phenomenological Theory of Martensite Crystallograpy (PTMC) by Bowles and
Mackenzie [6] as well as Wechsler, Lieberman and Read [7]. As a
phenomenological theory, PTMC simply links the initial and final states of the
martensitic transformation assuming a lattice-invariant shear mechanism.
Experimental determination of the habit plane was thus a necessity to validate
the PTMC results [8, 9]. While ground-breaking at the time, PTMC has since
been superseded by emerging theories that reproduce the experimentally
characterized habit planes with mechanistic models. Such models provide a
clear atomic path for martensitic transformation [10, 11, 12] and even predict
detailed properties of the interface [13].

Experimentally determining the habit plane trace is a challenging task in
itself. For example, plate martensite appears to thicken in both directions
away from the habit plane, whereas lath martensite thickens predominantly in
one direction [14]. Accordingly, the habit plane is sometimes approximated as
the average plane that contains the major circumference of the elongated child
grain [14], and at other times, as the flatter of the two long sides [15, 16].
The habit plane is conventionally defined by the three-dimensional orientation
of the martensitic plate in the austenite reference frame and is therefore
difficult to characterize directly. Early experimental studies [17, 18] relied
on the “two-surface technique” [19, 20] in which stereological analysis of
individual martensite plates was conducted in very large prior austenite
grains using two intersecting surfaces. Prior austenite orientations were
determined by stereologically observing the surrounding annealing twin
boundaries, or in case of sufficiently retained austenite, by means of the
back-reflection Laue method. The technique is limited since the orientation of
the martensitic laths or plates is missing. Despite these limitations,
Greninger and Troiano [18] observed plate-like martensite with an approximate
habit plane of $(225)_{\gamma}$ in a 1.40 wt.% carbon-iron alloy and
approximately $(259)_{\gamma}$ in a 1.78 wt.% carbon-iron alloy. Similarly,
Marder and Krauss [21] determined habit planes of approximately
$(557)_{\gamma}$ for packets of martensite laths in 0.2 and 0.6 wt.% carbon
steels.

With the advent of transmission electron microscopy, martensite morphologies
were studied at a much finer scale. The morphological observations were
further corroborated with orientation measurements from selected area electron
diffraction. Sandvik et al. [20] used thin films of individual martensitic
laths in an austenitic matrix in a Fe-20Mn-5Ni alloy to characterize the
crystallographic orientations of both lath and matrix. Upon tilting the thin
film to obtain an edge-on projection, the habit plane was obtained directly in
the austenite reference frame along with the identity of the specific variant
of the martensitic lath. The habit plane was determined as $(575)_{\gamma}$
from a total of 12 measurements. Without the edge-on projection, only the
trace of the habit plane can be observed in the imaging plane. When several
traces from differently oriented parent grains are measured, the habit plane
can be obtained by fitting a normal vector that is best orthogonal to all
traces. Using these fitting techniques, habit planes of lath martensite were
found to be near $(111)_{\gamma}$ [22],
4.5$\text{\,}\mathrm{\SIUnitSymbolDegree}$ from $(111)_{\gamma}$ [19, 22],
$(223)_{\gamma}$ [23] and $(557)_{\gamma}$ [16] for a number of steel alloys.

The limited number of past experimental studies that determined the habit
plane in martensitic steel indicates the considerable effort involved in such
work. Stereological analysis requires very large prior austenite grains and
cumbersome specimen preparation, while transmission electron microscopy
requires specimens with retained austenite. Both methods are labor intensive
and suffer from poor statistics even with optimal specimens. On the other
hand, scanning electron microscope -based orientation mapping via electron
backscattering diffraction (EBSD) is a fast and automated method to
characterize the morphology and crystallographic orientation of martensitic
structures in a planar section. Saylor et al. [24] determined the crystal
habits and the full five-parameter grain boundary character [25, 26] from
orientation maps. Furthermore, the method proposed by Saylor et al. [25] was
successfully applied to determine the intervariant grain boundary character
from bainite orientation maps [27]. While Ref.[24] may, in principle, be
suited to characterize habit planes in steels, the theory has three distinct
limitations: (i) the method relies on the presence of retained austenite,
which is not commonly present in martensitic steel microstructures, (ii)
martensitic variants are difficult to segment by conventional misorientation
angle thresholding [28], and (iii) all symmetry operations are applied to
rotate grain boundary segments from the specimen fixed reference frame into
the parent reference frame. The latter leads to problems in determining
martensitic habit planes (see Section 4.2).

With this motivation, we developed a new, extremely fast algorithm that
enables fully automated habit plane determination informed by parent grain
reconstruction on single planar cross-sections. We utilize reconstructed
parent phase grain maps to provide the parent reference frame and the
orientation relationship of the martensitic transformation. This allows habit
plane determination without the need for a retained parent phase and
furthermore, enables grain indexing at variant precision [28]. The application
of variant-specific orientation relationships allows us to determine the
dominant habit plane even from a single prior parent grain. We propose and
compare four distinct techniques for habit plane trace determination and
undertake a parameter sensitivity analysis that indicates exceptional
robustness of our method against poorly resolved child grains and distortions
in the orientation maps. The method delivers consistent results on orientation
maps of two mild steel alloys with 0.35 and 0.71 wt.% carbon and varying prior
austenite grain sizes, showing excellent agreement with habit planes reported
in literature. The method is openly available with MTEX 5.9 for the global
research community. All data and script files required for generating the
figures in this publication can be found at Ref. [29].

##  2 Theory

###  2.1 Mathematical description of habit planes

The parent-child interface at the atomic scale is closely related with the
operating orientation relationship. The habit plane is the macroscopically
visible interfacial plane between parent and child phases. An interface that
is planar and continuous at the atomic level directly corresponds to the habit
plane and can be derived from the crystallographic orientation relationship.
However, in the common case where the interface at the atomic scale consists
of regularly spaced ledges (Fig. 1), the habit plane becomes irrational, and
on a macroscopic level, detached from the crystallographic orientation
relationship [30, 31].

For the mathematical description of habit planes we consider two neighboring
grains with orientations $\mathbf{{O}}_{P\to X}$ and $\mathbf{{O}}_{C\to X}$
defined by the basis transformation from the crystal fixed reference frames of
the parent ($R_{P}$) and child ($R_{C}$) phases, respectively, to a specimen
fixed reference frame $R_{X}$. Accordingly, the misorientation between those
two orientations is the basis transformation

$\mathbf{{M}}_{P\to C}=\mathbf{{O}}_{C\to X}^{-1}\mathbf{{O}}_{P\to X}.$ (2.1)

that translates coordinates from the $R_{P}$ to the $R_{C}$ reference frame.

Figure 1: Projection of cubic parent and child lattices at their interface.
The ledge interface at the atomic scale is macroscopically visible as an
irrational plane. The latter is defined as the habit plane.

The orientations $\mathbf{{O}}_{C\to X}$, $\mathbf{{O}}_{P\to X}$ and their
resulting misorientation $\mathbf{{M}}_{P\to C}$ are subject to symmetry.
Considering the sets of symmetries $\mathcal{S}_{C}$ and $\mathcal{S}_{P}$ of
both phases, the orientations $\mathbf{{O}}_{C\to X}\mathbf{{S}}_{C}$ and
$\mathbf{{O}}_{P\to X}\mathbf{{S}}_{P}$ are physically indistinguishable for
all $\mathbf{{S}}_{C}\in\mathcal{S}_{C}$ and
$\mathbf{{S}}_{P}\in\mathcal{S}_{P}$. The misorientation $\mathbf{{M}}_{P\to
C}$ is subject to both sets of symmetries, $\mathbf{{S}}_{C}\mathbf{{M}}_{P\to
C}\mathbf{{S}}_{P}$ with $\mathbf{{S}}_{C}$ and $\mathbf{{S}}_{P}$ running
through all $C$ and $P$ symmetries, see Ref. [32].

The mathematical description of a habit plane is identical to the full five-
parameter description of a grain boundary [33]. We choose to describe a habit
plane by a triple $(\mathbf{{M}}_{P\to C}\mid\vec{n}_{P}\mid\vec{n}_{C})$
consisting of the misorientation $\mathbf{{M}}_{P\to C}$ and the plane normals
$\vec{n}_{P}$, $\vec{n}_{C}$ as coordinate vectors with respect to the
reference frames $R_{P}$ and $R_{C}$. For the habit plane to be valid, these
two coordinate vectors have to be consistent:

$\mathbf{{M}}_{P\to C}\vec{n}_{P}=\vec{n}_{C}.$ (2.2)

Analogous to the case of misorientations, the symmetry sets of the parent and
child phases produce symmetrically equivalent habit planes from a single habit
plane that cannot be physically distinguished:

$(\mathbf{{S}}_{C}\mathbf{{M}}_{P\to
C}\mathbf{{S}}_{P}^{-1}\mid\mathbf{{S}}_{P}\vec{n}_{P}\mid\mathbf{{S}}_{C}\vec{n}_{C})$
(2.3)

with $\mathbf{{S}}_{C}\in\mathcal{S}_{C}$ and
$\mathbf{{S}}_{P}\in\mathcal{S}_{P}$. The coordinates of habit plane normals
transform together with the specific choice of the misorientation. It follows
that if the misorientation is fixed, for e.g., in the fundamental region of
the misorientation space, the coordinates of the plane normals are fixed as
well.

A special case occurs if the misorientation $\mathbf{{M}}_{P\to C}$ allows for
symmetry operations $\mathbf{{S}}_{C}\in\mathcal{S}_{C}$ and
$\mathbf{{S}}_{P}\in\mathcal{S}_{P}$ such that

$\mathbf{{S}}_{C}\mathbf{{M}}_{P\to C}\mathbf{{S}}_{P}^{-1}=\mathbf{{M}}_{P\to
C}.$ (2.4)

In this case, the habit planes $(\mathbf{{M}}_{P\to
C}\mid\vec{n}_{P}\mid\vec{n}_{C})$ and $(\mathbf{{M}}_{P\to
C}\mid\mathbf{{S}}_{P}\vec{n}_{P}\mid\mathbf{{S}}_{C}\vec{n}_{C})$ are
symmetrically equivalent, have the same misorientation, but have different
plane normal coordinates. This phenomenon is known as the degeneracy of
variants [34] and is covered in more detail in our previous work [28]. Some
specific misorientations for which degeneracy can be observed are the Burgers
[35], Nishiyama-Wassermann [36] and Pitsch [37] orientation relationships.
Since the present study focuses on experimentally determined, irrational
orientation relationships [38], degeneracy should not occur and is therefore
not considered any further.

###  2.2 Habit plane traces

The algorithm developed in this study aims at determining a dominant habit
plane $(\mathbf{{M}}_{P\to C}\mid\vec{n}_{P}\mid\vec{n}_{C})$ from 2D
orientation maps of a single planar cross-section. We assume the orientation
relationship $\mathbf{{M}}_{P\to C}$ is known beforehand and the habit plane
normals $\vec{n}_{P}$ and $\vec{n}_{C}$ require determination. Since 2D data
is used, we are limited to observing habit plane traces formed by the
intersection of habit planes with the imaging plane only.

Considering two neighboring grains with given orientations $\mathbf{{O}}_{C\to
X}$ and $\mathbf{{O}}_{P\to X}$ that have an orientation relationship
$\mathbf{{M}}_{P\to C}$, the symmetry operations $\mathbf{{S}}_{C}$ and
$\mathbf{{S}}_{P}$ are obtained by the minimization of the misorientation
angle

$\omega\bigl{(}\mathbf{{S}}_{C}^{-1}\mathbf{{O}}_{C\to
X}^{-1}\mathbf{{O}}_{P\to X}\mathbf{{S}}_{P},\mathbf{{M}}_{P\to
C}\bigr{)}\to\min$

such that

$\mathbf{{O}}_{C\to X}^{-1}\mathbf{{O}}_{P\to
X}\approx\mathbf{{S}}_{C}\mathbf{{M}}_{P\to C}\mathbf{{S}}_{P}^{-1}.$ (2.5)

If the theoretical misorientation $\mathbf{{M}}_{P\to C}$ is far from a
degenerated orientation relationship and the experimental misorientation
$\mathbf{{O}}_{C\to X}^{-1}\mathbf{{O}}_{P\to X}$ is sufficiently close to
$\mathbf{{M}}_{P\to C}$, the symmetry operations $\mathbf{{S}}_{C}$ and
$\mathbf{{S}}_{P}$ are confidently determined.

Given that Equ. (2.5) is satisfied, the trace $\vec{t}_{X}$ of the habit plane
$(\mathbf{{M}}_{P\to C}\mid\vec{n}_{P}\mid\vec{n}_{C})$ observed on the
imaging plane with normal vector $\vec{n}_{S}$ is given by

$\vec{t}_{X}=(\mathbf{{O}}_{C\to
X}\mathbf{{S}}_{C}\vec{n}_{C})\times\vec{n}_{S}=(\mathbf{{O}}_{P\to
X}\mathbf{{S}}_{P}\vec{n}_{P})\times\vec{n}_{S}.$ (2.6)

In this equation, the habit plane trace $\vec{t}_{X}$ is observed in specimen
coordinates. Alternatively, the habit plane traces can be expressed in terms
of the coordinate vectors $\vec{t}_{C}$, $\vec{t}_{P}$ with respect to the
crystal reference frames $R_{C}$ or $R_{P}$:

$\vec{t}_{C}=\mathbf{{S}}_{C}^{-1}\mathbf{{O}}_{C\to X}^{-1}\vec{t}_{X}\text{
and }\vec{t}_{P}=\mathbf{{S}}_{P}^{-1}\mathbf{{O}}_{P\to X}^{-1}\vec{t}_{X}.$
(2.7)

After having employed Equ. (2.7), the habit plane normals $\vec{n}_{C}$ and
$\vec{n}_{P}$ may be obtained by finding the directions that are best
orthogonal to all determined traces $\vec{t}_{C}$ and $\vec{t}_{P}$,
respectively.

To further corroborate the crucial role of the correctly identified symmetry
operations, we plot a set of habit plane traces according to different
reference frames and symmetry operations (Fig. 2). Plotting the traces as
determined in the specimen fixed reference frame in a spherical projection
does not return useful information (see Fig. 2(a)). Transforming the traces
$\vec{t}_{X}$ from specimen into crystal coordinates by
$\vec{t}_{P}=\mathbf{{O}}_{P}^{-1}\vec{t}_{X}$ results in traces from common
prior parent grains (indicated by the same color) located on common circle
segments (Fig. 2(b)). These circles represent the imaging plane which has been
rotated into the parent reference frame for different prior parent
orientations. To find a single circle for all traces, we also need to consider
the symmetrically equivalent traces. However, considering all possible
symmetry operations to the traces is not sensible (Fig. 2(c)). On the other
hand, in Equ. (2.5) each variant is associated with a specific symmetry
operation $\mathbf{{S}}_{P}$. The application of this particular symmetry
operation in Equ. (2.7) yields traces in crystal coordinates that form a
common circle which is perpendicular to the habit plane normal (Fig. 2(d)). It
should be noted that in the case of degeneracy of variants, see Equ. (2.4), a
reduction towards a single circle is not possible as described in [39].

(a)

(b)

(c)

(d)

Figure 2: Stereographic projections of habit plane traces. (a) Pole figure of
trace directions in the specimen fixed reference frame. (b) Inverse polefigure
of traces that are natively rotated into the parent reference frame using the
experimentally determined orientations. (c) Inverse polefigure of all
symmetrically equivalent traces. (d) Inverse polefigure of traces transformed
with the symmetry operations determined from Equ. (2.5). The marker colors
distinguish variants formed within different parent grains.

##  3 Habit plane determination in fully transformed microstructures

###  3.1 General method

In the previous theory section, we have considered the case of two neighboring
parent and child grains. In actual orientation maps, it is common that fully
transformed microstructures do not contain any retained parent phase. With the
development and optimization of parent grain reconstruction in recent decades
[40, 28, 41, 42, 43, 44], we are able to determine the optimal average parent-
to-child orientation relationship $\mathbf{{M}}_{P\to C}$ which allows us to
reconstruct the prior parent orientation $\mathbf{{O}}_{P}$ from each child
orientation $\mathbf{{O}}_{C}$ with the assumption $\mathbf{{M}}_{P\to C}$. In
this instance, the child and prior parent orientation pairs
$(\mathbf{{O}}_{C}$ and $\mathbf{{O}}_{P})$ are equivalent to the neighboring
grains in Sections 2.1 and 2.2. Our algorithm for habit plane determination
requires the determination of the habit plane traces $\vec{t}_{X}$ which will
be discussed in Section 3.2. Following the transformation of traces from
specimen to crystal coordinates, the habit plane can be determined by finding
the direction that is best orthogonal to all determined traces.

Algorithm 1 summarizes our developed algorithm for the determination of the
dominant habit plane from an orientation map of a single planar cross-section.

######  Algorithm 1 (Habit plane determination in fully transformed
microstructures).

  1. (1)

Reconstructing child grains in the orientation map: This returns a list of
child orientations $\mathbf{{O}}_{C\to X}^{\ell}$, $\ell=1,\ldots,L$.

  2. (2)

Determining the best-fit orientation relationship $\mathbf{{M}}_{P\to C}$, for
example with one of the methods described in Refs. [14, 38, 40].

  3. (3)

Reconstructing parent grains: This returns a list of prior parent orientations
$\mathbf{{O}}_{P\to X}^{\ell}$, $\ell=1,\ldots,L$, corresponding to the child
orientations $\mathbf{{O}}_{C\to X}^{\ell}$.

  4. (4)

Determining the symmetry operations $\mathbf{{S}}_{C}^{\ell}$,
$\mathbf{{S}}_{P}^{\ell}$, $\ell=1,\ldots,L$ that minimize the misorientation
angle

$\omega\bigl{(}\mathbf{{O}}_{P\to
X}^{\ell}\mathbf{{S}}_{P}^{\ell},\mathbf{{O}}_{C\to
X}^{\ell}\mathbf{{S}}_{C}^{\ell}\mathbf{{M}}_{P\to C}\bigr{)}\to\min$

  5. (5)

Determining the habit plane traces $\vec{t}_{X}^{\,\ell}$, $\ell=1,\ldots,L$
of all child grains.

  6. (6)

Converting the traces into crystal coordinates using the specific symmetry
operations determined in step (4), i.e.,

$\vec{t}_{C}^{\,\ell}=(\mathbf{{S}}_{C}^{\ell})^{-1}(\mathbf{{O}}_{C\to
X}^{\ell})^{-1}\vec{t}_{X}^{\,\ell}\quad,\vec{t}_{P}^{\,\ell}=(\mathbf{{S}}_{P}^{\ell})^{-1}(\mathbf{{O}}_{P\to
X}^{\ell})^{-1}\vec{t}_{X}^{\,\ell}\,.$ (3.1)

  7. (7)

Determining the habit plane normals $\vec{n}_{C}$ and $\vec{n}_{P}$ as the
crystal directions best perpendicular to all traces $\vec{t}_{C}^{\,\ell}$ and
$\vec{t}_{P}^{\,\ell}$, $\ell=1,\dots,L$, respectively, as the solution of the
minimization problem

$\sum_{\ell=1}^{L}(\vec{\eta}_{C}\cdot\vec{t}_{C}^{\,\ell})^{2}\to\min,\quad\sum_{\ell=1}^{L}(\vec{\eta}_{P}\cdot\vec{t}_{P}^{\,\ell})^{2}\to\min\,.$
(3.2)

The exact solution of these minimization problems is provided by the
eigenvectors to the smallest eigenvalue of the matrices,

$\mathbf{{T}}_{C}=\sum_{l=1}^{L}\vec{t}_{C}^{\,\ell}\otimes\vec{t}_{C}^{\,\ell}\text{
and
}\mathbf{{T}}_{P}=\sum_{l=1}^{L}\vec{t}_{P}^{\,\ell}\otimes\vec{t}_{P}^{\,\ell},$

respectively. The robustness with respect to outliers is improved by first
solving this minimization problem for all traces, and then secondly, for only
80 percent of the data that is closest to the first solution.

Several methods to achieve steps 1-3 have been reported in the literature and
steps 4 and 3.1 were discussed in Section 2.2. Only step 5, the determination
of the habit plane traces remains. This is discussed in the following Section
3.2.

###  3.2 Trace determination

In theory, the trace of a habit plane is uniquely determined as the
intersection of the habit plane with the imaging plane. However, in
orientation maps, as considered in this study, the trace of the habit plane is
not necessarily directly identifiable. The habit plane trace in orientation
maps can only be approximated if the shape of the child grains is directly
related to the orientation of the habit plane. Oftentimes, the mechanical
constraints imposed on child grains during their nucleation and growth leads
to elongated grain shapes parallel to their habit plane. In this case, the
habit plane trace may be interpreted as the long axis of the grain or a
particular flat edge. In the following paragraphs, we propose four different
methods to determine the traces of elongated microstructure subsets and grains
which are interpreted as habit plane traces, suitable for most practical
applications. Two methods are pixel based and work directly with the EBSD maps
while the other two methods are vector based and operate on the polygonal
shape of the of grains.

#### Fourier transform based trace detection

Orientation data of the child phase contained within a single prior parent
grain is split into subsets of each orientation variant. Fig. 3(a) shows a
single reconstructed prior austenite grain of a martensitic steel
microstructure with orientation data from a red and blue martensitic variant.

(a)

(b)

(c)

Figure 3: (a) A red and blue variant within a single prior parent grain in
martensitic steel. The Fourier transform of (b) the blue and (c) red variants
are depicted together with the short axis of a fitted ellipse. The Fourier
transform was cropped to a circle to avoid bias during ellipse fitting toward
the edges of the rectangle.

The morphology in Fig. 3(a) is characterized by extended segments of constant
intensity along the long axis of the lamellar structure (i.e.- the trace) and
frequently oscillating intensity perpendicular to the long axis of the
lamellar structure. This directionally dependent oscillation in the binary
image becomes visible in the Fourier transform of the binary image of each
variant (Fig. 3(b) and Fig. 3(c)). The ellipse-shaped region of high intensity
is oriented perpendicular to the long axis of the lamellar structure. Using
principal component analysis of the Fourier transformed image, the long and
short -axes of the ellipses are determined to yield the traces. The Fourier-
based trace detection algorithm consists of the following steps:

######  Algorithm 2 (Fourier transform based trace determination).

  1. (1)

Generating binary images $X_{k,\ell}\in\\{0,1\\}$ for each variant within each
parent grain, with $1$ indicating the presence of the variant.

  2. (2)

Computing the absolute value of the Fourier transform
$Y=\left|\text{FFT}(X)\right|$

$Y_{m,n}=\left|\sum_{k}\sum_{\ell}X_{k,\ell}e^{2\pi i(km+\ell n)}\right|$
(3.3)

of the image $X$ and cropping of the largest possible circle, i.e. set
$Y_{m,n}=0$ for all $m,n$ with $m^{2}+n^{2}>\min(M^{2},N^{2})$.

  3. (3)

Principal component analysis on the Fourier transformed image $Y$, i.e.,
computing the trace direction $\vec{t}$ as the smallest eigenvalue of the
matrix

$\mathbf{{M}}=\begin{pmatrix}[c]\displaystyle\sum_{m,n}m^{2}Y_{m,n}^{2}&\displaystyle\sum_{m,n}mnY_{m,n}^{2}\\\
\displaystyle\sum_{m,n}mnY_{m,n}^{2}&\displaystyle\sum_{m,n}n^{2}Y_{m,n}^{2}\end{pmatrix}.$
(3.4)

Apart from the trace direction $\vec{t}$, the reliability index

$\varepsilon=\frac{\lambda_{\text{max}}-\lambda_{\text{min}}}{\lambda_{\text{max}}}$

from the matrix $\mathbf{{M}}$ is obtained which describes the roundness of
the ellipse-shaped region in Fourier space and varies between $0$ for a
perfect circle and $1$ for an infinitely stretched ellipse. A typical value
used as a threshold is $\varepsilon>0.2$.

#### Radon transform based trace detection

For a $K\times L$ image $X_{k,\ell}$ its Radon transform is a new image
$R_{m,n}$, the sinogram, where a pixel value $R_{m,n}$ is computed by summing
up all pixels $X_{k,\ell}$ along a line with angle $\omega_{m}$ and a shift
$n$. The angles are commonly chosen as
$\omega_{m}=0^{\circ},\ldots,180^{\circ}$ and the shifts as $n=0,\ldots,K+L$.

To apply the Radon transform for trace determination, the image $X_{k,\ell}$
is the binary image of all orientation data belonging to a specific variant of
a prior parent grain, which is identical to the initial image for the Fourier
transform method (Fig. 3(a)). The sinogram $R_{m,n}$ corresponding to the blue
variant in Fig. 3(a) is depicted in Fig. 4(a). Since the morphology of the
variant microstructures are characterized by long segments of constant
intensity parallel to the lamellar structure and frequently oscillating
intensity perpendicular to the lamellar structure, the sinogram is expected to
have large values for $\omega$ close to the trace direction that oscillates
with respect to the offset $n$. Indeed, Fig. 4(a) shows high oscillating
intensity values at $\omega\simeq 130$\text{\,}\mathrm{\SIUnitSymbolDegree}$$,
while the sinogram intensity is reduced and spread more evenly at
$\omega\simeq 40$\text{\,}\mathrm{\SIUnitSymbolDegree}$$.

The amount of oscillation at different angles $\omega_{m}$ can be quantified
by computing the Fourier transform through an FFT

$Y_{m,\nu}=\left|\sum_{n}R_{mn}e^{2\pi i\frac{n\cdot\nu}{K+L}}\right|$ (3.5)

of the sinogram $R_{m,n}$ with respect to the shift dimension $n$ (Fig. 4(b)).
The high frequency values of the Fourier transformed sinogram $Y_{m,\nu}$ are
confined to $\omega\simeq 130$\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and
evidently capture the characteristic spacing of the lamellar microstructure
subset. The low frequency values capture the overall shape of the prior parent
grain and are therefore not relevant to trace determination. The direction
$\omega_{m}$ with the largest intensity at high frequencies is obtained by
summing up the amplitudes $Y_{m,\nu}$ off all frequencies $\nu>\nu_{0}$ above
a certain threshold, e.g., $\nu_{0}=6$,

$Z_{m}=\sum_{\nu>\nu_{0}}Y_{m,\nu}.$ (3.6)

The angle $\omega_{m}$ corresponding to the largest value $Z_{m}$ is
considered as the trace direction. The detection of the maximum is made more
robust by applying a local smoothing filter to $Z_{m}$. Similar to Section
3.2, we use

$\varepsilon=\frac{\max Z_{m}-\min Z_{m}}{\max Z_{m}}$

as a quality measure for the determined trace direction. The Radon transform
based approach is summarized in Algorithm 3.

$0$$50$$100$$150$angle $\omega_{m}$ in degreeshift $n$ (a)

$0$$50$$100$$150$angle $\omega_{m}$ in degreefrequency $\nu$ (b)

$0$$50$$100$$150$angle $\omega_{m}$ in degree$Z(\omega)$ (c)

Figure 4: Steps of the Radon based trace determination method for the blue
variant in Fig. 3(a). (a) Sinogram $R_{m,n}$ of the binarized image
$X_{k,\ell}$. (b) Fourier transform $Y_{m,\nu}$ of the sinogram $R_{m,n}$ with
respect to dimension $n$. (c) Sum $Z_{m}$ of the FFT $Y_{m,\nu}$ as a function
of angle $\omega_{m}$.

######  Algorithm 3 (Radon transform based trace determination).

  1. (1)

Generating binary images $X_{k,\ell}\in\\{0,1\\}$ for each variant within each
parent grain, with $1$ indicating the presence of the variant.

  2. (2)

Computing the sinogram $R_{mn}$ as the Radon transform of the binary image
$X_{k,\ell}$ with respect to angles $\omega_{m}$ and shifts $n$.

  3. (3)

Computing the Fourier transform $Y_{m,\nu}$ of sinogram $R_{mn}$ along the
second dimension by Equ. (3.5).

  4. (4)

Summing the Fourier transformed sinogram $Y_{m,\nu}$ for all frequencies $\nu$
above a threshold $\nu_{0}$ by Equ. (3.6) resulting in a direction depended
magnitude $Z_{m}$.

  5. (5)

Determining the angle $\omega_{m}$ corresponding to the largest value in
$Z_{m}$ as the trace direction.

#### Boundary segment based trace determination

While the two previous methods relied on orientation data of individual
variants within prior parent grains, the third method works on boundary
segments of grains. The first step is reconstructing the child grains from
orientation maps and extracting of all grain boundary segments as a list of
angles $\beta_{k}$ and lengths $\ell_{k}$, see e.g. [45]. The strategy
identifies traces as the dominant directions of boundary segments.

(a) (b) (c) (d)

Figure 5: Grain boundaries and the circular histograms of the grain boundary
segment angles $\beta_{k}$ (a,b) before and (c,d) after grain boundary
smoothing.

Due to the regular grids in orientation maps, the boundary angles $\beta_{k}$
are typically multiples of $60$\text{\,}\mathrm{\SIUnitSymbolDegree}$$ or
$90$\text{\,}\mathrm{\SIUnitSymbolDegree}$$, yielding an angular distribution
in a circular histogram (Fig. 5(a) and 5(b)) that is not representative of the
grain boundary orientations in the microstructure. This so-called staircase
effect is reduced by smoothing the grain boundaries, i.e., moving grain
vertices such that their energy is minimized while preserving triple points.
Grain boundary smoothing considerably dampens the staircase effect (Fig. 5(c)
and 5(d)) with respect to the raw data, even though a slight preference of the
$0$\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and
$90$\text{\,}\mathrm{\SIUnitSymbolDegree}$$ angles remains.

Considering a lamellar morphology of the child phase, we expect more and
longer boundary segments parallel to the long direction of the lamellar
structure with respect to the orthogonal direction. The circular histogram of
all boundary segments associated with the grains of the individual variants in
Fig. 6(b) supports this notion. Computing the joint circular histogram for all
child grains of a specific variant and prior parent grain rather than for each
child grain separately drastically improves the robustness of this method. The
robustness is further improved by employing instead of the circular histogram
a circular kernel density estimate [46]:

$f(\beta)=\sum_{j}e^{-\frac{\sigma^{2}j^{2}}{2}}\sum_{k}\ell_{k}e^{ij(\beta_{k}-\beta)}.$
(3.7)

The resulting density estimate is visualized by the curves that encircle the
histograms in Fig. 6(b). Its shape depends on the bandwidth or smoothing
parameter $\sigma$. In the present study, a constant value of
$\sigma=10$\text{\,}\mathrm{\SIUnitSymbolDegree}$$ gave satisfactory results.
However, data-driven bandwidth selection methods may be employed as well [46].
Finally, the trace direction is estimated by the modal angle of the density
function $f$:

$\omega=\operatorname*{argmax}_{\beta}f(\beta).$

The trace determination algorithm is summarized as follows:

######  Algorithm 4 (Circular histogram based trace determination).

  1. (1)

Identifying all boundary segments of a specific variant belonging to a common
prior parent grain.

  2. (2)

Computing their lengths $\ell_{k}$ and angles $\beta_{k}$.

  3. (3)

Computing the circular kernel density estimator $f(\beta)$ according to Equ.
(3.7) with a suitable smoothing parameter $\sigma$.

  4. (4)

Determining the trace direction $\omega$ as the modal value of the function
$f$.

(a) (b)

(c)

Figure 6: (a) Grains associated with two specific child variants in a prior
parent grain and two traces determined from (b) the modal direction of the
fitted density function (solid curves) from a circular histogram plot of the
boundary directions, and (c) the characteristic shapes of the previously
considered child variants.

A more robust method for analyzing the preferred orientation of a set of
grains is using their characteristic shape, see Ref. [47]. Analogous to the
previous approach, the characteristic shape -based algorithm works with a list
of angles $\beta_{k}$ and lengths $\ell_{k}$ of the grain boundary segments
and identifies traces as the dominant directions of boundary segments. To
derive the characteristic shape of a set of grains, the boundary segments,
ordered according to their angle $\beta_{k}$, are concatenated. Since all
grains have closed boundaries, rearranging their boundary segments must result
in a closed polyhedron. Ordering grain boundary segments according to their
angle results in similar angles for neighboring grain boundary segments; thus
yielding a characteristic shape in the form of a convex and smooth polyhedron.
Fig. 6(c) shows the characteristic shapes of all blue and red variant grains.
The long axis direction of the characteristic shape, computed as its largest
eigenvalue, is interpreted as the trace. Compared to the circular histogram
based approach (Algorithm 4), the characteristic shape based approach is
robust against the staircase effect and therefore requires significantly less
smoothing. The algorithm is summarized as follows:

######  Algorithm 5 (Characteristic shape based trace determination).

  1. (1)

Identifying all boundary segments of a specific variant belonging to a common
prior parent grain.

  2. (2)

Computing the characteristic shape of these boundary segments according to
Ref. [47].

  3. (3)

Computing the trace direction $\omega$ as the long axis of the characteristic
shape.

##  4 Method validation and application

Algorithm validation comprises sensitivity analysis of the different
parameters involved in habit plane determination as well as habit plane
determination in a synthetic microstructure with a known habit plane. We then
apply and validate our algorithm on real lath martensitic steel
microstructures with systematically varied martensite morphology, prior
austenite grain size and magnification and compare our findings with those
from literature.

###  4.1 Sensitivity to deviating trace directions

There is a common preconception that accurate habit plane determination from
orientation maps is impossible, as orientation mapping techniques suffer from
distortions of the orientation map and specifically EBSD suffers from
insufficient spatial resolution [48, 49, 50]. It is therefore essential to
test the robustness of our general algorithm against correlated and
uncorrelated rotations of the trace directions in the imaging plane.
Uncorrelated rotations may be associated with low spatial resolution or an
irregular grain morphology, whereas correlated rotations describe possible
distortions originating from _(i)_ electron beam drift, _(ii)_ low
magnification and _(iii)_ angular deviations between the assumed imaging plane
and the actual sample surface. _(i)_ and _(ii)_ may be addressed by good
practice in orientation data acquisition and distortion correction during
post-processing. For _(iii)_ however, even small deviations in specimen tilt
may introduce large rhomboidal distortions [48] that may be difficult to avoid
during routine measurements.

(a)

(b)

(c)

Figure 7: (a) Normal rotation distributions with a standard deviation of
12$\text{\,}\mathrm{\SIUnitSymbolDegree}$ and a median value of 0 (blue) and
10$\text{\,}\mathrm{\SIUnitSymbolDegree}$ (orange), (b) Rotated traces
applying the blue distribution and (c) the orange distribution from panel (a).
The quantiles of angular deviations from the determined trace are shown for 25
%, median and 75 %, alongside the angular deviation angle between actual and
determined habit plane.

Poor child grain resolution and map distortions are reflected in a rotation of
the actual habit plane trace around the normal vector of the imaging plane.
Therefore, we define two normal distributions of rotation angles with a
standard deviation of 12$\text{\,}\mathrm{\SIUnitSymbolDegree}$ around the
image plane normal vector (Fig. 7(a)). The first distribution is centered
around 0$\text{\,}\mathrm{\SIUnitSymbolDegree}$, simulating poor child grain
resolution, while the second is centered around
10$\text{\,}\mathrm{\SIUnitSymbolDegree}$, simulating an additional effect of
a distorted orientation map. We generated a synthetic dataset comprising 150
random parent orientations in a cubic reference frame and a given $(575)_{P}$
habit plane in the cubic parent reference frame. Applying Equ.2.6, the habit
plane traces were derived from the given habit plane and parent grain
orientations. After applying the rotation distributions in Fig. 7(a) to the
habit plane traces, the habit planes were determined following the steps 3.1
and 3.2 in Algorithm 1. Applying rotational noise to the trace direction to
simulate poor child grain resolution led to a negligible angular deviation
from the actual habit plane of 0.3$\text{\,}\mathrm{\SIUnitSymbolDegree}$
(Fig. 7(b)). Strikingly, even adding the additional effect of a distorted
orientation map only led to an angular deviation of
0.4$\text{\,}\mathrm{\SIUnitSymbolDegree}$ (Fig. 7(c)). The pole figures
indicate that any type of rotation applied to the habit plane traces in the
imaging plane increases the variance of the habit plane traces in the parent
reference frame. Following our algorithm, the rotation of the measured habit
plane traces into the parent reference frame (Algorithm 1, Step 3.1)
introduces an angular deviation between the measured trace and the actual
habit plane. Here the application of different parent grain orientations and
symmetry operations causes a randomization of the systematic offset introduced
by the distortion from the orange distribution in Fig. 7(a). The grain
statistics provided by state-of-the-art orientation maps therefore render our
algorithm robust against low accuracy in the trace determination and
distortions in the orientation map.

(a)

(b)

(c)

Figure 8: Pole figures showing habit plane traces from the sensitivity
analysis of misindexed symmetry operations (a) A fourth of the parent
orientations were misindexed as their twinned orientations. (b) A poor
representative orientation relationship led to symmetry operation misindexing.
(c) Superposition of the effect of imperfectly measured traces from Fig. 7(b)
and the effect of misindexed symmetry operations in panel (b). The true habit
plane, (575), is indicated by the red line.

###  4.2 Sensitivity to misindexed symmetry operations

The rotation of habit plane traces from the specimen fixed reference frame
into the parent reference frame (Algorithm 1, Step 3.1) requires knowledge of
the parent grain orientations. In our algorithm, the parent orientations are
determined via parent grain reconstruction (Algorithm 1, Step 3). The
reconstruction of wrong parent orientations leads to wrongly identified (i.e.-
misindexed) symmetry operations in Step 4 of Algorithm 1. We therefore conduct
a sensitivity analysis for two prominent sources of misindexing: _(i)_
Reconstruction of the parent twin orientation caused by shared variants
between a parent grain and its twin, which is particularly prominent in lath
martensitic steels [41], and _(ii)_ Parent grain reconstruction assuming a
non-representative orientation relationship (Algorithm 1, Step 3).

For sensitivity analysis in case _(i)_ , we consider that 10 percent of the
perfectly identified habit plane traces are mistakenly brought into the parent
reference frame by the symmetry operations of the parent twin (Algorithm 1
Step 4). Fig. 8(a) shows that the false symmetry operations generate a second
erroneous habit plane, which shares the $[\overline{1}01]_{P}$ direction with
the true habit plane.

To analyze case _(ii)_ , we depart from the Greninger-Troiano orientation
relationship [4] to compute child grain orientations and expose these to
random rotations of maximum 10$\text{\,}\mathrm{\SIUnitSymbolDegree}$ to
reflect local variance in the orientation relationship. Then we obtain the
symmetry operations via the Kurdjumov-Sachs orientation relationship [51]
(Algorithm 1 Step 4), which is a non-representative orientation relationship
for the considered parent and child grains. Fig. 8(b) shows the resulting
habit plane traces under the influence of misindexed symmetry operations.
While the actual habit plane is most prominent, several alternative habit
planes are suggested. The second most prominent habit plane is $(755)_{P}$.
Following the notation by Morito et al. [52], this is the habit plane
associated with variant 4, which is the variant with the lowest misorientation
to variant 1. Since these variants tend to form adjacent to each other within
lath martensite blocks, this particular misindexing is likely to be observed
in experimental data. In contrast to twin misindexing in case _(i)_ , the two
most prominent habit planes cross each other at $[55\overline{12}]_{P}$,
rather than $[\overline{1}01]_{P}$.

The final case for sensitivity analysis in Fig. 8(c) is a superposition of the
effects of randomly rotated trace directions from Fig. 7(b) and the example of
a non-representative orientation relationship in Fig. 8(b). Despite the
significant simulated errors that are purposely introduced in Steps 4 and 5 of
Algorithm 1, the determined habit plane shows an angular deviation of only
2.8$\text{\,}\mathrm{\SIUnitSymbolDegree}$ from the true habit plane. We
conclude that, while our algorithm is extremely robust against variance in the
trace directions, it is more sensitive to inaccurate parent grain
reconstruction. Error in the latter may propagate into the determination of
wrongly identified symmetry operations in step 4 of our algorithm.
Nevertheless, even in quite extreme cases, the habit plane is confidently
determined within less than 3$\text{\,}\mathrm{\SIUnitSymbolDegree}$ of
angular deviation.

###  4.3 Validation on an artificial orientation map

To validate the developed algorithm on a microstructure with a known habit
plane, we produced a synthetic parent-child microstructure with a resolution
of $640^{3}$ voxels using Dream3D [53] (Fig. 9(a)). The parent phase comprised
equiaxed grains, while the child phase was made up of plates of varying
thickness (Fig. 9(b)). Both phases were assigned a cubic symmetry, a
consistent orientation relationship and a $(255)_{P}$ habit plane. We
extracted a 2D orientation map with $640^{2}$ pixel resolution from the 3D
microstructure, fitted the orientation relationship (Algorithm 1 Step 2) and
applied neighbor-level voting-based parent grain reconstruction [28] (Fig.
9(c), Algorithm 1 Step 3). For the habit plane trace determination in Step 5
of Algorithm 1 we applied the Algorithms 2 \- 5, all of which returned 82
habit plane traces (traces given by Algorithm 3 shown in Fig. 9(b)).

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9: (a) Synthetic microstructure consisting of a lamellar cubic child
phase with habit plane $(255)_{P}$ embedded in a retained cubic parent phase.
(b) Extracted 2D map showing parent and child grains and the determined habit
plane traces. (c) Reconstructed parent phase orientation map. (d) Inverse pole
figure comparing the actual $(255)_{P}$ habit plane with the determined habit
planes. (e) Magnified region from panel (d) showing the deviation from the
true habit plane. (f) Magnified region from panel (b) showing deviations in
the determined traces.

Fig. 9(d) shows the determined habit planes vs. the actual $(255)_{P}$ habit
plane in an inverse pole figure. The Radon, characteristic shape and circular
histogram methods determine the habit planes accurately, the first with no
deviation and the latter two deviating only
$0.3\text{\,}\mathrm{\SIUnitSymbolDegree}$ from $(255)_{P}$. The Fourier
method gives the poorest result, deviating
$1.7\text{\,}\mathrm{\SIUnitSymbolDegree}$ from the true habit plane. The
origin of this minor deviation becomes apparent when examining the individual
traces within the microstructure in Fig. 9(f): the Fourier method introduces a
clear angular deviation to the traces. It is apparent that the Fourier method
is biased towards the Feret diameter of the longest direction of each child
grain in the present microstructure. All methods still perform excellently
considering the small sample size of only 82 traces.

###  4.4 Validation and application on experimental orientation maps

Lean carbon steels containing 0.35 wt.% and 0.71 wt.% carbon (hereafter
referred to as compositions 035C and 071C) were each annealed at different
temperatures to produce distinct austenite ($\gamma$) grain sizes and fully
martensitic ($\alpha^{\prime}$) microstructures during quenching. We acquired
orientation maps and reconstructed the prior austenite microstructure to
enable determination of the habit planes following Algorithm 1 with the traces
acquired in Step 5 using the methods presented in Section 3.2 (Algorithms 2 \-
5). Details on specimen preparation, thermal treatments, data acquisition and
post-processing are provided in Appendix 1. The annealing temperatures and
resulting prior austenite grain sizes are given in Table 1. In the table,
condition A denotes a lower annealing temperature compared to condition B. The
grain sizes are reported in Table 1 as the mean equivalent circle diameter
(ECD) of grains with an ECD exceeding $1\text{\,}\mathrm{\SIUnitSymbolMicro
m}$. For condition 071C-B only an approximate lower bound for the prior
austenite grains is stated as the prior austenite grain boundaries extended
over the boundaries of the orientation maps.

Figure 10: Pole figures showing density functions for the rotated traces for the specimen 0.71C-A. A) an example of the traces transformed to a density function and plotted on a pole figure, b) the Fourier method, c) the Radon method, d) the histogram method and e) the characteristic shape method. The habit plane is depicted as a red circle and the habit plane normal as a red square. $[\overline{1}01]$ is marked with a white square. Alloy | ID | Anneal |  ECDγ ($\text{\,}\mathrm{\SIUnitSymbolMicro m}$) | Algorithm | Traces | Habit |  $\overline{\theta}_{n_{p}\rightarrow t_{p}}$ ($\text{\,}\mathrm{\SIUnitSymbolDegree}$) |  Runtime (s)
---|---|---|---|---|---|---|---|---
035C | A |  $870\text{\,}\mathrm{\SIUnitSymbolCelsius}$ 5 min | 4 | Fourier | 16769 | 9 10 9 | 14.5 | 149
Radon | 16769 | 11 12 11 | 11.0 | 107
Histogram | 16801 | 11 12 11 | 11.4 | 71
Shape | 16801 | 11 12 11 | 9.5 | 40
B |  $1050\text{\,}\mathrm{\SIUnitSymbolCelsius}$ 1 h | 17 | Fourier | 4011 | 9 11 9 | 12.4 | 53
Radon | 4011 | 7 8 7 | 8.8 | 82
Histogram | 4017 | 8 9 8 | 8.8 | 27
Shape | 4017 | 7 8 7 | 7.5 | 32
071C | A |  $820\text{\,}\mathrm{\SIUnitSymbolCelsius}$ 5 min | 8 | Fourier | 9780 | 8 12 9 | 10.4 | 93
Radon | 9780 | 8 12 9 | 7.5 | 117
Histogram | 9775 | 11 12 11 | 7.3 | 46
Shape | 9775 | 11 12 11 | 5.5 | 31
B |  $1050\text{\,}\mathrm{\SIUnitSymbolCelsius}$ 1 h | $>$40 | Fourier | 3183 | 3 4 3 | 12.6 | 97
Radon | 3183 | 3 4 3 | 7.7 | 143
Histogram | 3188 | 11 12 11 | 10.8 | 48
Shape | 3188 | 11 12 11 | 7.2 | 40
Table 1: Alloy designations of the 0.35 wt.% and 0.71 wt.% carbon steels with
their annealing conditions and average parent austenite grain sizes alongside
the results from habit plane determination utilizing different trace
determination methods (Algorithms 2 \- 5). Results show the number of
determined traces, the approximated habit plane, the mean angular deviation
between traces and the habit plane as well as the runtime for Algorithms 2 \-
5

(a)

(b)

(c)

(d)

Figure 11: A section of a variant map (a,c) and reconstructed prior austenite
map (b,d) for specimens (a,b) 0.35C-A and (c,d) 0.35C-B. Annealing twin
boundaries have been highlighted with magenta (a,c) and cyan (b,d).

(a)

(b)

(c)

(d)

Figure 12: A section of a variant map (a,c) and reconstructed parent austenite
map (b,d) for the specimens (a,b) 0.71C-A and (c,d) 0.71C-B. Annealing twin
boundaries have been highlighted with magenta (a,c) and cyan (b,d).

.

Figure 10 shows examples of density functions calculated from the rotated
traces for specimen 071C-A. In all cases, the traces are located around a
distinct band in the parent reference frame, making up the habit plane.
Depending on the algorithm used in step 5 to identify the traces, different
levels of scatter around the habit plane can be observed in the Figure. Table
1 gives the scatter for each method as mean angle of deviation between the
traces and the habit plane, $\overline{\theta}_{n_{p}\rightarrow t_{p}}$.
Judging from the mean deviations, the Radon (Algorithm 3) and the
characteristic shape (Algorithm 5) methods result in the most reliable traces.
Conversely, the Fourier method (Algorithm 2) resulted in the greatest scatter
in all cases. Table 1 also shows the number of detected traces for each
specimen and method alongside with the approximate habit plane, as well as
examples of the combined runtimes of Algorithms 2 \- 5 for each specimen, when
run on a system with an Intel(R) Core(TM) i5-10310U processor and 16 GB
physical memory. The algorithms are quite fast and do not put any constraints
on the size of the analyzed datasets.

Figures 11 and 12 show representative examples of the acquired martensitic and
the corresponding reconstructed austenite microstructures. The increase in
annealing temperature has expectantly increased the prior austenite grain size
in both compositions. Moreover, for both annealing conditions, the prior
austenite grain size of the 071C composition is significantly larger than for
the 035C composition. The figures show martensitic grains with a lath
morphology, colored on the basis of the martensitic variant number according
to Ref. [52]. As is typical for lath martensite, neighboring laths are
observed to share the same packets. Some exceptions to this general
observation are noted in condition 071C-B, which appears to have formed some
larger martensite units at the grain boundaries of the parent austenite. This
morphology may be associated with the early nucleation and growth of
martensite at the parent austenite grain boundaries as observed in Ref. [54].

(a)

(b)

Figure 13: The dominant habit planes shown in the austenite reference frame
(a) a standard inverse pole figure and (b) a magnified inset of panel (a). The
angular deviation between the Radon and the characteristic shape algorithms
for trace determination are indicated and habit planes $(575)_{\gamma}$ and
$(252)_{\gamma}$ are shown as references from previous studies on lath
martensite.

Fig. 13 shows the habit planes projected on an inverse pole figure. Habit
planes of the 035C composition are close to $(111)_{\gamma}$, while the 071C
composition shows a habit plane closer to $(575)_{\gamma}$. We follow the
habit plane notation by Sandvik et al. [16], in which the habit plane is
reported in the austenite reference frame related to variant 1 according to
Ref. [52] for the Kurdjumov-Sachs orientation relationship [51] as
$(111)_{\gamma}||(011)_{\alpha^{\prime}}$ and
$[\overline{1}01]_{\gamma}||[\overline{1}\overline{1}1]_{\alpha^{\prime}}$.

As mentioned in the introduction, studies on martensite habit planes have in
later years mainly focused on TEM work conducted on special alloys with
isolated martensite laths in an austenitic matrix. Comparative data for lean
steel alloys is provided by the two-surface trace studies by Greninger and
Troiano [18], as well as Krauss and Marder [21]. The former mainly reported
results for steel compositions exceeding 1.0 wt.% carbon, noting the scarcity
of reliable observations for lower-carbon steels. The latter employed the two-
surface trace method on a 0.2C and a 0.6C iron-carbon alloy, prepared by
annealing in an evacuated quartz capsule at approximately
$1050\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 1h and then breaking the
capsule in water. The specimen preparation is similar for 035C-B and 071C-B. A
habit plane for 0.6C close to $(575)_{\gamma}$ was determined by observing
aggregated lath traces from a single packet of martensite. This result, when
comparing the habit planes on an inverse pole figure, almost exactly overlaps
the result obtained for the large-grained 071C-B, shown in Fig. 13. Marder and
Krauss show similar results for the 0.2C composition, reporting a habit plane
in the vicinity of $(575)_{\gamma}$. This is in disagreement with the current
study, where the habit plane for a slightly higher carbon concentration is
shown to be in the vicinity of $(111)_{\gamma}$. Marder and Krauss noted that
their observations suffered from the difficult specimen preparation and
extreme scarcity of observations typical to two-surface studies. Other works
have indicated habit planes in the vicinity of $(111)_{\gamma}$ for low-carbon
lean steels [18, 55] and other alloys with lath martensite [22]. The good
agreement between the Radon and characteristic shape methods for trace
determination in this study, as well as the similarity in results between the
two different grain sizes and orientation map resolutions corroborate that the
habit plane in 0.35C lean steel is closer to $(111)_{\gamma}$ than
$(575)_{\gamma}$.

Fig. 13(b) indicates that the habit planes computed from trace determination
using Radon, histogram and characteristic shape -based methods are very close
to each other for the 035C specimens. The different methods result in more
scatter for the 071C-specimens, with the largest angular deviation of
$2.9\text{\,}\mathrm{\SIUnitSymbolDegree}$ between the characteristic shape
and the radon -based algorithms. Fig. 10 shows the density functions (half-
width $5\text{\,}\mathrm{\SIUnitSymbolDegree}$) of the determined traces for
specimen 071C-A (with the largest scatter between the results) in the parent
reference frame. For all algorithms, the density of traces forms a distinct
band perpendicular to the determined habit plane normal, with a prevalent mode
at approximately $[\overline{1}01]_{\gamma}$. The sensitivity analysis in
Section 4.2 revealed that this mode at $[\overline{1}01]_{\gamma}$ may be
caused by misindexed symmetry operations on account of parent austenite
erroneously reconstructed with its parent twin orientation. The reconstruction
of annealing twins in austenite is a known issue [41, 43] making this a likely
source of the $[\overline{1}01]_{\gamma}$ mode. Alternatively, there might be
a physical explanation for the mode at $[\overline{1}01]_{\gamma}$. Sandvik
and Wayman [16] reported that the long axis of laths is $\langle
110\rangle_{\gamma}$, which could indicate that the observed mode reflects a
probability distribution of observing elongated grains at different crystal
directions in the imaging planes. The origin of the mode will be the subject
of future research.

While no definitive conclusion is reached to explain the scatter in the
results for condition 071C-A, a possible reason might be the distinct
martensite morphology at the prior austenite grain boundaries. The
characteristic shape and the histogram algorithms are grain boundary -based
methods and therefore sensitive to irregular grain shapes, potentially making
them vulnerable to a local change in martensite morphology. The observation
that the deviation is largest at the smaller grain size corroborates this
suspicion, as a higher density of parent austenite grain boundaries would
strengthen this source of error. However, seeing past these minor differences
introduced by the trace determination methods, both habit planes for 071C-A
are clearly in the vicinity of $(575)_{\gamma}$ and similar to 071C-B.

##  5 Conclusions

We developed and validated a new algorithm for the fully automated
determination of the dominant habit plane from orientation maps of a single
planar cross-section. The algorithm couples the five-parameter grain boundary
character with the habit plane and the orientation relationship between parent
and child phases. Parent grain reconstruction is used to obtain the prior
parent orientations and to identify the symmetry operations required for the
transformation of the habit plane traces from a specimen fixed reference frame
to a parent or child reference frame. In addition to the development of the
main algorithm, we also proposed four different and powerful algorithms for
the automated detection of habit plane traces. Extensive testing of the
algorithm revealed the following findings:

  1. (1)

The algorithm is extremely robust against random and systematic deviations in
the trace angle, representing the scenarios of poor child grain resolution and
distortion of the orientation map. The robustness is based on the statistics
of traces associated with different child variants from several prior parent
grains.

  2. (2)

The algorithm is extremely fast.

  3. (3)

The algorithm is more sensitive to inaccuracies in parent grain
reconstruction. Imperfect parent grain reconstruction may cause wrong symmetry
operations to be identified. This, in turn, propagates to an incorrect
transformation of traces from the specimen to crystal reference. For
martensitic steel, the common case of erroneous parent grain reconstruction of
a twin-related parent orientation was investigated and led to the suggestion
of an alternative habit plane that intersected the actual habit plane at
$[\overline{1}01]_{\gamma}$. Also, misindexed symmetry operations due to a
non-representative orientation relationship had similar, although less severe,
effects with a common intersection of habit planes at
$[55\overline{12}]_{\gamma}$.

  4. (4)

Tests on a synthetic microstructure with known habit plane successfully
validate our algorithm. It also highlighted minor characteristic differences
in the choice of method for trace determination.

  5. (5)

Tests on martensitic steel microstructures with different characteristic
martensite morphologies and habit planes at different prior austenite grain
sizes and orientation map resolutions resulted in consistently determined
habit planes that corroborate previous results from literature obtained using
other techniques. In this case, accurately reconstructed parent austenite was
key to correct habit plane determination.

The automated method was implemented in conjunction with the parent grain
reconstruction features in MTEX 5.9 and is freely and publicly available to
the scientific community. All data and script files required for generating
the figures in this publication can be found at Ref. [29].

## Acknowledgements

Professor Bevis Hutchinson is gratefully acknowledged for fruitful discussion
on the subject of habit planes and for his comments on the manuscript.

## Appendices

### 1\. Experimental methods

The composition of the steel alloys were Fe-0.35C-0.38Mn-0.14Cr and
0.71C-0.21Si-0.5Mn-0.09Cr-0.13Ni wt.%. Both alloys were in sheet form and
exposed to two different heat treatments to produce different austenite grain
sizes. The first heat treatment consisted of austenitization for 5 minutes at
approximately Ae3 + $70\text{\,}\mathrm{\SIUnitSymbolCelsius}$ according to
the iron-carbon phase diagram, followed by quenching in saturated brine. The
second heat treatment consisted of exposing the samples sealed in evacuated
quartz capsules to $1050\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 1h,
followed by breaking the capsule on the inside of a steel water container,
which was then vigorously stirred.

The specimens were cross-sectioned with a laboratory disc cutter and hot
mounted ($180\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 540s ) in conductive
resin, ground and polished in stages, and finished by polishing for 420s with
colloidal silica in order to obtain a deformation-free surface for EBSD
characterization.

A Zeiss Gemini 450 field emission gun - scanning electron microscope with an
Oxford Instruments Symmetry CMOS detector was used to acquire the orientation
maps. Three orientation maps were obtained for each specimen, with dimensions
of approximately 275 by 206 µm at a step size of 0.15 µm. An exception was
made for the specimen 0.71C-B due to its very large parent austenite grain
size. For this specimen, six maps were measured with dimensions approximately
367 by 275 µm, at a step size of 0.2 µm. Each map was obtained from the
quarter thickness region of the cross-section.

Post processing of the orientation map data was carried out in MTEX 5.8.2 in
conjunction with Matlab R2022a. Prior to reconstruction and habit plane
determination, single-pixel grains were removed from the map. A grain map was
then constructed from the orientation map, using an angular threshold of
3$\text{\,}\mathrm{\SIUnitSymbolDegree}$. A representative orientation
relationship was determined for each dataset, using the iterative algorithm of
the reconstruction suite in MTEX [38]. The parent austenite grain map was then
reconstructed using the two-step version of the variant graph algorithm [41].
In the first step, the threshold
$\delta=$2.5\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and tolerance values
$\sigma=$2.5\text{\,}\mathrm{\SIUnitSymbolDegree}$$ were used to create the
variant graph, followed by merging the close-together variants. The graph was
then run through five iterations of the algorithm. The variant-level precision
was restored in the second step, followed by two additional rounds of
iteration. This directly gave the parent orientations for each grain in the
map, which was then used to reconstruct the parent orientation map on a pixel
level, with a best-fitting variant index of the representative orientation
relationship assigned to each pixel. In addition, each pixel was assigned the
index of its corresponding parent austenite grain.

The original child orientation map was then re-segmented based on the variant
level information. Instead of using an angular threshold, each pixel was
considered to be part of the same sub-domain when it shared the same parent
grain and same OR variant index as its neighbors. The variant-level grain map
was then smoothed to lessen the staircase effect.

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