# Modeling of experimentally observed topological defects inside bulk
polycrystals

Siddharth Singha, He Liub, Rajat Arorac, Robert M. Suterb, Amit Acharyad a
Department of Civil and Environmental Engineering, Carnegie Mellon University,
5000 Forbes Avenue, Pittsburgh, PA 15213 USA  b Department of Physics,
Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213 USA
cAdvanced Micro Devices, Inc. (AMD), 7171 Southwest Pkwy, Austin, TX 78735 USA
dDepartment of Civil and Environmental Engineering and Center for Nonlinear
Analysis, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 15213
USA<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A rigorous methodology is developed for computing elastic fields generated by
experimentally observed defect structures within grains in a polycrystal that
has undergone tensile extension. An example application is made using a near-
field High Energy X-ray Diffraction Microscope measurement of a zirconium
sample that underwent $13.6\%$ tensile extension from an initially well-
annealed state. (Sub)grain boundary features are identified with apparent
disclination line defects in them. The elastic fields of these features
identified from the experiment are calculated.

††: Modelling Simulation Mater. Sci. Eng.

, ,

Keywords: Polycrystal, topological defects, high energy diffraction
microscopy, elasticity

##  1 Introduction

The recent development of three dimensional x-ray microscopies that use unit
cell orientations as the contrast mechanism [1, 2, 3, 4, 5, 6] has opened up
the possibility of making rigorous comparisons between observed and modeled
features and responses in interior volumes of metal and ceramic
polycrystalline microstructures. For example, comparisons have been made
between observed plastic behaviors and model expectations including twinning
[7], grain rotation and breakup [8, 9] as well as crack nucleation and
propagation [10, 11, 12]. Studies of thermal responses include
recrystallization [13, 14, 15] coarsening [16, 17, 18, 19] and phase
transformations leading to abnormal grain growth. [20] These studies provide
direct experimental observations that can and are being used to develop
improved models of materials responses. For example, experimental evidence
shows that long held assumptions with respect to coarsening are inconsistent
with observations based on x-ray microscopy. [16, 19]

Here, a demonstration is presented in which an experimentally observed
topologically interesting feature is observed in the orientation field in a
near-field high energy diffraction microscopy (nf-HEDM) measurement in the
interior of a crystalline grain of zirconium after tensile deformation. The
elastic distortion field around this feature is modeled (as a two dimensional
system) to show the possible implication of an underlying discontinuity in the
orientation field on the stress fields generated in a purely elastic medium.
To our knowledge, there do not exist research efforts linking experimental
measurements of orientation distributions to the possible stress fields
associated with them. Restricting the discussion to the linear theory of
elasticity, it is a fundamental result that Nye tensor fields arising from
skew symmetric tensor fields cannot produce stress. However, terminating
rotation discontinuities or discontinuities with varying magnitude of rotation
jump across them can produce stress corresponding to distributions of
disclinations and disclination dipoles in such “interfaces.” The dipoles are
topologically equivalent to interfacial dislocations. We leverage this idea to
identify, at the scale of the observations, plausible intra-granular
orientation discontinuities and use g.disclination theory and associated
computational techniques in two dimensions to compute their stress fields [21,
22]. Related work can be found in [23, 24] focusing on defect kinematics where
finite difference approximations are used for inferring the disclination
density from observations, but no stress fields are calculated. The work in
[25] utilizes grain averaged elastic strain measurements to compute
intergranular elastic distortion and stress fields in a polycrystal using
field dislocation mechanics theory (a subset of our equations), constraining
the computed elastic strain field’s grain-wise averaged field to equal the
measured grain averaged elastic strain field.

##  2 Experimental methods and sensitivity

High energy diffraction microscopy (HEDM) is described in detail in [6] which
also summarizes a variety of applications. Here, salient aspects of the data
collection procedures and the reconstruction algorithms used here for near-
field HEDM (nf-HEDM) measurements are pointed out. While the data collection
is time consuming and reconstructions are computationally intensive, the
method outlined here is quantitative, has high dynamic range in grain size and
is sensitive to subtle features. For example, the method allows the mapping of
small angle intra-granular orientation variations and, as a result, the
identification of interesting topological features such as those of interest
here.

###  2.1 Near-field HEDM measurement

The implementation at the Advanced Photon Source of the near-field HEDM (nf-
HEDM) method [4, 5, 6] is based on high resolution ($\sim 1.5\ \um$) imaging
of diffracted beams that emerge from illuminated grain cross-sections as a
polycrystalline sample rotates in the line focused monochromatic x-ray beam.
During rotation, each grain that is illuminated by the quasi-planar beam
passes through Bragg conditions corresponding to a set of Miller indices,
$\\{h,k,l\\}$, or, equivalently, reciprocal lattice vectors, $\bf{G}_{hkl}$.
If the grains are “good” crystals (i.e., have low defect content and therefore
sharp Bragg peaks), then each volume element of each grain’s cross-section
satisfies Bragg conditions simultaneously (at the same values of rotation
angle, $\omega$). Each illuminated grain volume element produces a diffracted
beam in the direction of the outgoing wavevector,
$\bf{k}_{f}=\bf{k}_{i}+\bf{G}_{hkl}$, with $\bfk_{i}$ being the incident beam
wavevector. These parallel beams generate anisotropically projected images of
horizontal grain cross-sections on a two dimensional detector oriented
perpendicular to the incident beam. Diffracted beam images are collected as
the sample rotates (about an axis perpendicular to the plane defined by the
incident beam) through contiguous intervals of width
$\delta\omega\leq\degrees{1}$ as the sample rotates through a total range
$\Delta\omega=\degrees{180}$. To encode the directions of each scattered beam,
measurements are done at two rotation axis-to-detector distances, $L_{1}$ and
$L_{2}$. The sets of projected images from each grain allow information about
grain shapes, location and crystal unit cell orientations to be deduced by
reconstruction software.

If grains have substantial defect content and are therefore not “good”
crystals, they may have orientation gradients which will spread diffracted
beams over arcs on the detector images and over multiple neighboring
$\delta\omega$ intervals. In cases where grains break up into discrete blocks,
for example due to deformation via single slip, the above description still
holds if we re-define what we mean by a ”grain:” a grain becomes any region in
the sample over which the unit cell orientation is uniform on the scale of the
orientation resolution of the measurement. Continuous lattice bending can lead
to intensity patterns in which individual pixels receive contributions from
sample regions with different orientations; such patterns become difficult to
quantitatively interpret. A further complication arises from the fact that the
intensity striking a given pixel arises from a sample volume determined by the
scattering geometry specified by the orientation of $\bfk_{f}$ and the
scattering angles conventionally referred to as $2\theta$, with $\theta$ being
the Bragg angle, and an azimuthal angle, $\eta$; these angles are determined
by the orientation and magnitude of $\bfG_{hkl}$ for each Bragg peak.
Nevertheless, nf-HEDM has been shown to at least statistically track
intragranular orientation variations. [26]

The above experimental procedure yields a set of $N=2\
\Delta\omega/\delta\omega$ detector images that encode a two dimensional unit
cell orientation field in the illuminated layer that generated them. Full
three dimensional information is obtained by measuring successive layers as
the sample is translated normal to the incident beam plane. Since x-rays are
non-destructive in hard materials (metals, ceramics, minerals), successive
measurements can be carried out between processing steps that alter the
microstructure to reveal heterogeneous responses. For example, evolution under
heat treatments [16, 17, 18, 27, 14] and mechanical loading [7, 8, 9, 28] have
been carried out using this near-field method. [6]

###  2.2 NF-HEDM Reconstructions: sensitivity to intragranular lattice
rotations

Converting a measured set of diffraction images into a discretized map of the
microstructure that produced them is referred to as a reconstruction. For the
present, the “microstructure” refers only to the field of unit cell
orientations (additional quantities such as material density, phase, [20]
composition, and elastic strain state [29] can be added) and the output is
frequently referred to as a “grain map.” Assuming the line focused x-ray beam
has a thickness well below the grain size, the approximation is made that the
cross-section is a two dimensional plane through the sample. A sample
coordinate system [4] is defined as having its $z_{s}$-axis along the $\omega$
rotation axis and perpendicular to the plane of the beam (typically vertical
in the laboratory); the $x_{s}$ and $y_{s}$ axes are both defined as
perpendicular to $z_{s}$ and parallel and perpendicular to $\bfk_{i}$,
respectively, when the rotation angle $\omega=0$; these axes form a right
handed coordinate system. The origin of $z_{s}$ is in the incident beam plane
and $x_{s}$, $y_{s}$ origins are defined to be at the rotation axis. Note that
many other conventions are used in various laboratories and in other software
packages.

The Python-GPU based HEXOMAP package [30] is used to perform reconstructions
for this work. [31] The code uses a C++ library of routines that were written
for the well exercised and verified [32] previous generation code. [5]
Preliminary image analysis routines (incorporated in HEXOMAP) extract signal
from background in the diffraction images as described in [33]. HEXOMAP
optimizes orientations on a square array of points in the illuminated sample
space. It is straightforward to import three dimensional sets of these arrays
into a variety of analysis and materials simulation codes. Orientations are
reported as $R(\bfr_{s})$ where $R$ represents an active rotation that takes
the reference unit cell orientation and rotates it into the orientation
determined at sample position, $\bfr_{s}$.

A brief summary of the orientation optimization process will help to justify
the sensitivity to intra-granular features used here. Given the experimental
geometry and a trial unit cell orientation at a particular sample point or
voxel position, $\bfr_{s}$ (in the illuminated plane at $z_{s}$), the software
determines the value of $\omega$ at which each Bragg peak will occur, computes
the laboratory position of the sample point at that $\omega$, and projects the
diffracted beam from that position, along $\bfk_{f}$, and onto the detectors.
If the diffracted beam strikes detector pixels that contain intensity in the
experimental measurement, then an overlap counter is incremented. A confidence
or completeness metric, $\mathcal{C}$, is computed as the ratio of the number
of peaks that should strike the detector and overlap experimentally observed
signal to the total number of simulated peaks that should do so. The
optimization routine performs a hierarchical search over a symmetry reduced
fundamental zone of possible orientations and reports back the orientation
that maximizes $\mathcal{C}$. The orientation in each voxel undergoes a final
local Monte Carlo optimization that removes any discreteness due to the
orientation mesh.

In the nf-HEDM geometry, over the $\Delta\omega=\degrees{180}$ rotation range,
and for arbitrary unit cell orientations, $R$, typical elemental metallic
crystals generate 50 to over 100 Bragg peaks that strike detectors. The
optimal $R$ for each voxel puts each of its peaks into the correct sample
rotation interval of width $\delta\omega$ and makes each peak overlap
experimentally observed spots within those images (with two detector
placements measuring the same intervals, the simulation must overlap
experiment at both detector positions). The orientation is highly constrained.
Empirically, using $\delta\omega=\degrees{1}$ results in orientation noise
within “perfect” crystals of $\leq\degrees{0.1}$. [32] Unit cell rotations
through angles well below $\delta\omega$ will move a subset of peaks that
occur near the edges of the orientation bins into neighboring intervals. Thus,
even small unit cell rotations deep within grains (where projections should
land in the interiors of experimental spots) can shift intensity to
independently measured images.

The computationally intensive procedure outlined above combined with
relatively slow layer-by-layer data collection makes these quantitative
measurements significantly more expensive than other grain mapping methods
that also resolve intra-granular properties. [34, 35] However, the 2D-to-2D
projection of sample cross-sections to area detectors makes feature signals
transparently accessible (rather than having to be extracted from intensity
profiles) and the voxel-by-voxel reconstruction assumes only local, micron
scale crystallinity. In this paper, we take advantage of these aspects of nf-
HEDM and HEXOMAP to resolve interesting internal defect structures within the
grains of an deformed zirconium polycrystal.

###  2.3 Measurement of tensile deformation of zirconium

The data set that is investigated here is reported in [7, 36]. Detector images
span $\delta\omega=\degrees{1}$ integration intervals. A custom built loading
system that is compatible with the near-field experimental geometry applies
tensile load under displacement control with a load cell reading the applied
force. The nf-HEDM measurement includes data from 50 layers separated by $4\
\um$ in $z_{s}$ in each of four extension states, denoted S0 through S3. The
focus here is on states S0, the pristine annealed state with well ordered
grains and state S2 in which tensile extension of $13.6\ \%$ has been applied.

The original analysis, using IceNine reconstructions, [5] focuses on
deformation twinning observations [7] that run counter to expectations based
on the sample’s “clock” texture with $c$-axes predominantly almost
perpendicular to the loading axis. An example of pyramidal slip induced
lattice rotations (which are favored by the texture) is also observed. [36]
The sample’s cross-sectional area evolved anisotropically, again due to
texture. The $c$-axes of grains are not only preferentially oriented far from
the loading axis, they are also clustered in the perpendicular plane. In S0,
the average sample cross-section dimension is $883\ \um$ and in S2 it is $825\
\um$.

The results reported here are based on new HEXOMAP reconstructions of states
S0 and S2 using the original raw experimental image data. The reconstructions
are qualitatively similar to those in [7]; quantitative comparisons cannot be
carried out since the older reconstruction output files were lost in a disk
crash. Each layer reconstruction was performed at sample points on $1.48\
\um\times 1.48\ \um$ meshes spanning the sample cross-section dimensions.
Because the detector pixel pitch is $1.48\ \um$, this mesh is at the
measurement resolution of individual diffraction images.

##  3 Experimental observations

Fig. 1 shows a single cross-section through the zirconium sample’s
microstructure in the S0 and S2 states. This cross-section is in the center of
the measured volume: It is layer number 25 of 50 in S0 and number 27 of 50 in
S2. In both cases, only voxels with $\mathcal{C}>0.4$ are plotted. In (a), the
great majority of voxels are shown in orientation coded colors while in (b),
extended low confidence areas having $\mathcal{C}<0.4$ (black) indicate severe
deformation and/or substantial elastic strain. In both images, black lines are
drawn between all neighboring reconstructed points with $\mathcal{C}>0.4$ that
have orientations that differ by a rotation angle greater than 1.1. In (a),
only well defined grains are delineated again indicating well defined
crystalline grains. In (b), grain boundaries are still observed and many
grains that appear in (a) remain, albeit with deformed shapes, but a
substantial population of intra-granular misorientations ($\Delta$) with
discontinuities $>\degrees{1.1}$ also appear. The green grain marked with a
“$*$” is the subject of the analysis below (see Fig. 7).

(a)[scale=.48]figures/Layer_S0-star.png
(b)[scale=.48]figures/Layer_S2-star.png

Figure 1: Corresponding cross-sections through the microstructure in (a) S0
and (b) S2 states. Axis units are in millimeters and colors are mapped from
the three parameters specifying unit cell orientations (Rodrigues vector
components) The grain of interest is marked with a red ’*’ in both states.

The grain of interest seen in Fig. 1 is shown as extracted from its
surrounding grains in state S2 in Fig. 2 which also demonstrates that the
orientation field in this grain is relatively uniform, having no
discontinuities larger than $\degrees{5}$ (panel (a)). But, well defined
discontinuity lines are seen in (b) with the lower threshold of 1.1. This
verifies the small angle nature of all interior boundaries and the relative
uniformity of orientations around them. The line of orientation
discontinuities just below the * in Fig. 1b is the feature whose associated
stress field is considered below.

(a) [scale=.45]figures/Grain_wo_D.png (b)
[scale=.45]figures/Grain_W_defect.png
(c) [scale=.45]figures/Gradient_feat.png

Figure 2: The extracted grain cross-section chosen for analysis in state S2.
The black boundary lines are drawn for (a) misorientation
($\Delta)>\degrees{5}$ and (b) $\Delta>\degrees{1.1}$ as in Fig. 1.(c) Shows
how the value of $\Delta$ varies along the feature of interest. The
orientation based color variations in (a) and (b) are difficult to distinguish
by eye.

##  4 Modeling of the observed orientation field

Our goal here is to demonstrate a procedure to estimate the linear elastic
stress fields associated with what are variously referred to as low angle,
small angle or sub-grain boundaries as they are observed in HEDM data. We
denote these boundaries as s-gbs. Such s-gbs are usually thought of as being
comprised of a dislocation wall but, noting that dislocations can be
decomposed into disclination dipoles, we adopt the language of disclinations.
Individual disclinations store high levels of elastic energy in the far-field
region away from the defect core and therefore usually appear in dipole
combinations. However, our calculations are sensitive to longitudinal
variations along boundaries of the rotation jump across the boundary. An
extreme example of such a variation is if the boundary terminates within a
grain; a termination corresponds topologically to a bare disclination.
Similarly, less extreme variations observed along the boundary correspond to a
continuous distribution of disclination density.

All the defects discussed above occur on the nano-scale and are not explicitly
observed in the nf-HEDM measurements. However, the longer length scale
consequences of collections of defects yield the unit cell rotations that are
observed at experimental resolution. We surmise, as is commonly done, that
nano-scale defect arrays underlie the s-gbs. The computations yield stress
fields on the meso-scale length scales of the measurements.

In this section, we describe our interpretation of the experimental data,
introduce the computational model and describe the stress fields obtained from
the computations.

###  4.1 Interpretation of voxelized experimental data

Based on the measurements, we say that an s-gb lies in the jagged region
formed by the voxels across which an above threshold orientation discontinuity
lies as illustrated in Fig. 3.

(a) [scale=.55]figures/Jagged_sgb_exp.png (b)
[scale=.55]figures/Jagged_sgb.png

Figure 3:  Outline of s-gb identification for computations: (a) Black dots are
points at which the reconstruction software determines rotation values
(lattice orientations). The analysis associates rotation values with the area
of the corresponding square box. Two different rotations are shown on either
side of an actual s-gb (bold black line). The stepped red line represents the
s-gb as identified by the measurement. (b) For modeling, we identify boxes
(marked as light blue) across which there exist large rotation gradients. We
obtain the s-gb shape by passing a best fit low order polynomial (dashed black
line) through the centers of border voxels.

However, we attribute a smooth shape and a normal to that interface by
connecting the voxel centers abutting it which correspond to regions of high
gradients in measured orientations (described by the magnitude of the field
$P$ subsequently defined in (11)). The smoothed interface so defined is
assumed to be the surface/curve across which the orientation jumps. In this
interpretation of the data, no orientation (or distortion) can be assigned to
the interface itself – and the tangential gradient of orientation/distortion
is assumed to vanish at the interface. Thus, the only measurement attributable
to the interface is an assertion of the normal gradient of orientation across
it, given by the jump in orientation/distortion, divided by the width of the
boundary, an input to the theory. This normal gradient field constitutes the
eigenwall field, $S$, of the theory defined in Sec. 4.2. The theory suggests
that any region of the interface where the orientation/distortion jump varies
longitudinally (that is, along the dashed line in Fig. 3(b) along it serves as
a source for stress (a location of non-zero $\mathit{\Pi}$ field). For
example, in a twin boundary with curvature, this must be true as the
distortion jump depends on the normal. We note that a tangential variation, of
the jump in orientation normal to the interface, is not related in any way to
assigning orientations to the interface, or a tangential gradient of the
orientation field on the interface. In our calculations, we will be
conservative and assign the width of the boundary to be a voxel length, having
the effect of predicting lower stresses due to the identified defects. The
calculations in this paper involve linear partial differential equations
(pde), so this amounts to a scaling of stress fields by a constant factor.

###  4.2 Elements of g.disclination theory

We review here the static elastic g.disclination theory of [21, 22]. The
(inverse) elastic distortion and the eigenwall fields are the main kinematic
ingredients of the model. Unlike interfaces of (inverse) displacement
discontinuity which may or may not be physically identifiable (e.g.,
identifiable stacking faults, or ‘perfect’ 1-atom spacing shifts across an
atomic layer otherwise), (inverse) elastic distortion discontinuities (at a
macroscopic scale) are always identifiable. If $W$ denotes the (inverse)
elastic distortion, at an interface like a phase or grain boundary it has a
large gradient over a thin interfacial region; we think of this concentrated
gradient as the ‘singular part’ of $W$ and refer to it as the eigenwall field
$-S$. The singular part is separated out of the (distributional) derivative of
$W$, $DW$, to define the ‘regular part’ of $DW$ as

$Y:=DW-(-S)=DW+S.$ (1)

$Y,DW,S$ are third order tensor fields.

When the elastic distortion $U^{e}$ is small, $W\approx I-U^{e}$, in which
case (1) can be expressed as

$Y=-DU^{e}+S.$

With $X$ as the alternating tensor and the definition $Y:X=:\alpha$,

$Y:X-S:X=\mbox{curl}\,U^{e}\qquad\Longrightarrow\qquad\alpha+(-S:X)=\mbox{curl}\,U^{e}.$
(2)

Since both terms on the l.h.s. of (2) are sources of incompatibility of the
elastic distortion (since $\mbox{curl}\,U^{e}=0$ implies that $U^{e}$ is a
gradient in simply connected domains), (2), along with the physical
interpretation of the eigenwall field, says that $-S:X$ is the _interfacial
dislocation density_ and $\alpha$ is the _bulk dislocation density_ field.
Both serve as sources for elastic distortion of the body causing stress (in
many, but not all, circumstances).

Another important source of elastic distortion is the _g.disclination density_
$\mathit{\Pi}$ defined as

$\mbox{curl}\,Y=\mbox{curl}\,S=:\mathit{\Pi}.$

In the following we define $\mathit{\Omega}$ as the domain of interest in
which we want to calculate elastic distortion fields and stress due to
interfacial dislocation density fields (and possibly bulk dislocation density
fields). As explained in [22], in order to deal with stress fields arising
from longitudinal gradients of the eigenwall field along grain and compatible
phase boundaries, i.e., g.disclination density fields, we define

$\displaystyle\eqalign{\mbox{curl}\,{S}^{\perp}&=\mbox{curl}\,{S}\qquad\mbox{on}\
\mathit{\Omega}\\\ \mbox{div}\,{S}^{\perp}&=0\qquad\mbox{on}\
\mathit{\Omega}\\\ {S}^{\perp}n&=0\qquad\mbox{on}\ \partial\mathit{\Omega}\\\
-\mbox{div}\,D{H}&=\mbox{div}\,{S}\qquad\mbox{on}\ \mathit{\Omega}\\\
(-D{H})n&={S}n\qquad\mbox{on}\ \partial\mathit{\Omega}.}$

so that

${S}={S}^{\perp}-D{H}.$

Then, with the definition $\hat{U}^{e}:=U^{e}+{H}$, (1) may be equivalently
stated as

$Y-{S}^{\perp}=-D\hat{U}^{e}$ (4)

so that

$\alpha-{S}^{\perp}:X=\mbox{curl}\left(U^{e}+{H}\right)=:\mbox{curl}\,\hat{U}^{e}.$

Finally, defining the constitutive equation for stress to be
$\sigma=\mathbbm{C}\,\hat{U}^{e}$, where $\mathbbm{C}$ is the tensor of
(possibly anisotropic) linear elastic moduli, the elastostatic field equations
at ‘small distortions’ become

$\displaystyle\eqalign{\alpha-{S}^{\perp}:X&=\mbox{curl}\,\hat{U}^{e}\\\
\mbox{div}(\mathbbm{C}\,\hat{U}^{e})&=0\qquad\mbox{on}\ \mathit{\Omega}\\\
(\mathbbm{C}\,\hat{U}^{e})n&=t\qquad\mbox{on}\ \partial\mathit{\Omega},}$ (8)

where $t$ is a statically admissible applied traction field (possibly
vanishing).

The l.h.s. of (8) shows the sources of elastic distortion. It can be shown
[22, Sec. 6] that when $S$ is skew in its first two indices (as happens with
the modeling of low angle s-gbs, as is the case here) the elastic strain and
stress depend only on the knowledge of the bulk dislocation density $\alpha$
and the g.disclination density $\mathit{\Pi}$. It can also be inferred that,
under the same assumption, if $S^{\perp}$ is replaced by $S$ and $\hat{U}^{e}$
by $U^{e}$ in (8), then $\hat{U}^{e}_{sym}=U^{e}_{sym}$ and the stress and
elastic strain remain unchanged. However, the rotation fields
$\hat{U}^{e}_{skw}\neq U^{e}_{skw}$, if $S\neq S^{\perp}$.

###  4.3 Computation of elastic fields of grain and sub-grain boundaries

(a) [scale=.45]figures/model_gb.png (b) [scale=.45]figures/model_sgb.png
(c) [scale=.45]figures/model_comb.png

Figure 4: Misorientation (in degrees, $\degrees{}$) is presented for (a) the
network of grain boundaries, (b)the s-gb, (c) the combined system for which
the stress field is calculated. Different colorbar scales are used to make
features more evident. Note that upper limit of colorbar in (c) is restricted
to $\degrees{2.5}$.

In this section, we discuss the approach adopted in this paper to calculate
the elastic fields of an interfacial network comprising grain boundaries
(high-angle boundaries) and s-gb (low-angle boundaries). Grain boundaries have
a large angle of misorientation in comparison to s-gbs within grains, and form
a continuous network without terminations. On the other hand, unlike grain
boundaries, s-gbs may be accompanied by high gradients of misorientation in
the longitudinal direction along the feature, as evident in Fig. 5.

[scale=.50]figures/log_alpha_norm.png

Figure 5: Interfacial dislocation density showing the network of grain
boundaries; $\alpha^{(i)}$ is defined in A. For measurements with satisfactory
certainty, the (terminations in the) s-gb are not captured.

Significant longitudinal gradients along a gb can occur only at junctions
where another gb meets it, almost by definition. If these transverse gbs are
not taken into account, the large longitudinal gradients along a gb
necessarily produce large stresses, as the ‘sites’ of such longitudinal
gradients serve as (continuously distributed) disclinations that act as
sources of elastic distortion with non-trivial strain. For a realistic
prediction of stresses within and around a single grain due to interfacial
defect fields, it is thus important to account for the network of gbs
connected to a single closed gb. Moreover, if only rotation can be measured,
and the presence of strain can make the measurement uncertain, then for higher
degree of confidence (defined in Sec. 3) the grain boundary network may not
appear connected as shown in Fig. 2(c). For this practical reason related to
modeling with experimental data, we reduce the confidence value of
reconstruction used in Fig. 2(c) set at 0.4 to 0.2 to obtain the gb network
shown in Fig. 4(a). The corresponding stress field of the gb network is
discussed at the end of Sec. 4.5.

The theoretical framework employed in this first paper for computing stress of
experimentally measured interfacial features as outlined in Sec. 4.2 applies
to the small elastic distortion case (the theory and calculations in [22] are
not so restricted). However, the presence of a grain boundary, as discussed,
implies large misorientations and hence large elastic rotations, and the
utilization of the aforementioned linear theory for calculation of stress is
not appropriate. As a compromise, instead of using a nonlinear elastic
constitutive relation for stress and an interfacial dislocation density field
calculated from a finite rotation tensor field with large misorientation
representing the gb network, we define an infinitesimal rotation tensor field
$\omega^{e}:=X\Phi$ from the measured finite rotation vector field $\Phi$ (a
vector with magnitude representing the angle of rotation in radians and
direction giving the axis of rotation) and calculate its stress field;
clearly, $\mbox{curl}\,\omega^{e}$ concentrates on the gb network and it
produces small stresses as would the fully nonlinear theory without
approximation, while identifying the gb network accurately. As a validation of
our approach, we have plotted the $\log_{10}$ of the norm of the interfacial
dislocation density (Fig. 5) obtained from this $\omega^{e}$ field and we note
that grain boundaries are observed quite well.

For the s-gb, high gradients in the interfacial dislocation density along the
s-gb is observed in the experimental data (Fig. 2) and consequently should
show up in the modeling. An s-gb, by definition, comprises a ‘small’
misorientation and, with the additional (self-consistent) assumption that the
elastic strain field produced by such s-gb features is small (which,
nevertheless, can produce stresses that are two to three orders of magnitude
higher than typical macroscopic yield stresses), the linear elastic theory
discussed in Sec. 4.2 can be used to compute its stress field. Then, the
overall stress field of the gb network combined with the s-gb is obtained by
superposition due to the use of linear elasticity for calculation of stresses.

Mathematically, let $F=R\sqrt{C}$ represent the unique pointwise polar
decomposition of the (finite) elastic distortion tensor into an orthogonal
elastic rotation tensor $R$ and the right stretch tensor $\sqrt{C}$, and we
define $\varepsilon:=\sqrt{C}-I$. Then the inverse elastic distortion tensor
field is $W=(I+\varepsilon)^{-1}R^{T}$. In any given situation, the actual
strain field $\varepsilon$ is non-vanishing in general, but we assume that
only the rotation field $R$ is measured. We now assume that across an (s-)gb
with normal $n$ the rotation field can appear discontinuous with values
$R_{t},R_{b}$ while the strain field is continuous. Then across such an
(s-)gb, the difference in elastic distortion is given by

$\Delta
W=(I+\varepsilon)^{-1}\left(R_{t}^{T}-R_{b}^{T}\right)\approx\left(R_{t}^{T}-R_{b}^{T}\right)-\varepsilon\left(R_{t}^{T}-R_{b}^{T}\right)$

with the approximation valid when $|\varepsilon|\ll 1$. In such a case the
source of elastic distortion, including the elastic strain, according to
g.disclination theory is $S:X$ with $S$ well approximated by

$S=\left(R_{t}^{T}-R_{b}^{T}\right)\otimes n,$

and we have the interesting result that observations of the (finite) rotation
field alone, without measurement of strain, can provide the source for
internal stress to a good approximation in many situations of physical
relevance. The magnitude of this source is small for an s-gb, and hence the
use of the linear theory for calculation of stress is justified. For a gb,
this magnitude of the eigenwall field is large and the linear theory for
computing stress is not justified, and we resort to the device mentioned in
the previous paragraphs.

Two remarks are in order here: first, the computational implementation of
$\Delta W$, the field $P$ in Sec. 4.4, succeeds in identifying (s-)gbs with
terminations that are in reasonable accord with those in the experimentally
observed misorientation field, where the latter does not involve any construct
of g.disclination theory, see Figs. 7 (a) and 2 (c). Second, the result in A
implies that if the linearized formula $X:\Phi$ instead of $\exp(X:\Phi)$
corresponding to a finite rotation vector field $\Phi$ is used to define the
rotation field $R$ utilized above, it would not be possible to quantitatively
identify terminating interfaces and their stress fields (as measured by the
$\mbox{curl}\,\alpha^{T}$ field).

###  4.4 Definition of the input to the computational model

In this first demonstration, we restrict ourselves to calculations of elastic
fields that vary only in two dimensions and, of course, we choose the measured
plane discussed above in Sec. 3. For modeling, the restriction to the plane
requires the definition of rotations with axes perpendicular to the chosen
plane. In the following, we describe the procedure for defining this input. We
note that the restriction to two dimensions is only for simplicity - our
theory and experimental techniques are appropriate for 3-d analysis and finite
deformation elastic field analysis.

The theory works with the inverse elastic distortion tensor, $W$, and we
assume that discontinuities in inverse elastic distortion occur only through
its rotation field. Thus, we focus on the measured orientation field whose
transpose gives the rotation of the inverse elastic distortion. To convert the
$3\times 3$ rotation matrix data from the measurement for our 2-d analysis, we
retain the upper-left $2\times 2$ sub-matrix of the $3\times 3$ transpose of
the rotation matrix written in the sample basis. The distortion field, $W$, so
obtained is not a pure rotation field, but this is not an impediment for our
elastic field calculations as it can treat full distortion discontinuities and
not just rotational ones.

With the 2-d distortion field in hand, we define a field $P$ as a discrete
gradient of $W$:

$\displaystyle\eqalign{DW_{i}(x)&=W(x+he_{i})-W(x-he_{i})\\\
P&=\sum_{i=1}^{2}DW_{i}\otimes e_{i}.}$ (11)

$P$ captures the idea of how $W$ varies in all directions. The calculation of
$P$ is illustrated in Fig. 6. The value of $P$ in the gray voxel is calculated
from the difference in the 2-d distortion field values in the adjacent blue
voxels in two orthogonal directions.

[scale=.48]figures/P_cal_m_up.png

Figure 6: Calculating $P$ field for the gray voxel using distortion data
available on the blue voxels.

$P$ contains information about the discontinuity in $W$ at the voxel level. To
characterize the strength of discontinuities, we define the magnitude of $P$
as $|P|=\sqrt{P_{ijk}P_{ijk}}$. Fig. 7 shows maps of $|P|$ in the selected
grain shown in Fig. 2. For comparison, $|P|$ is shown based on both the 2-d
and 3-d rotation specifications. In both cases, internal s-gbs appear as
expected from Fig. 2. Smaller values of $|P|$ also appear frequently as
continuous features.

[scale=.48]figures/P_norm_2d.png (-7.5,5)(a)
[scale=.48]figures/P_norm_3d.png(-7.5,5)(b)

Figure 7:  Color map of $|P|$ for the selected grain of interest. For
comparison, (a) shows $|P|$ for the reduced 2-d (inverse) elastic distortion
field while (b) shows $|P|$ for the full 3-d rotation field. Scale: 1 unit =
1.48$\um$.

To define the distortion field at surrounding grain boundaries, distortion
data corresponding to voxels having low confidence values ($\mathcal{C}$ in
the reconstruction) were used. Low $|P|$ magnitude along some parts of the gb
is attributed to the fact that the low $\mathcal{C}$ value orientations in
voxels around those parts is nevertheless similar to that inside the grain. As
discussed above, these voxels are not included in the grain definition because
of their low confidence value. It is also observed that $|P|$ is smaller for
the s-gb when compared to gb as is expected. Discontinuities that appear in
the s-gb in the 2-d case that are less obvious in the 3-d case, are most
likely because of changes in the dominant rotation axis. Low-level features
are prevalent in 3-d cases because full rotations are not restricted to the
single, perpendicular axis.

####  4.4.1 Construction of the eigenwall field from experimental data

With the observation that s-gbs have larger $|P|$ than background features, we
set a threshold significantly above the background level and extract voxels
with values of $|P|$ above the threshold. These voxels are then used to define
s-gbs. For the simulations, only a single s-gb feature, shown in Fig. 8, is
used to simulate the elastic fields. Given the smoothed s-gb shape, we next
calculate the value of the distortion jump, or eigenwall field $S$, to be
assigned to each voxel.

[scale=.48]figures/wo_padding.png (-8,5)(a) [scale=.49]figures/w_padding.png
(-8,5)(b)

Figure 8:  Simulation domain containing (a) just the s-gb with a prescribed
Eigenwall field. (b) s-gb and network of grain boundaries having interfacial
dislocation density. The elastic computational domain is shown in blue.

The eigenwall field $S$ takes third-order tensors as field values and provides
a discrete directional derivative of the orientation, in the direction of the
normal, $n$, to the s-gb. $n$ is not constant for the curved s-gbs. It is
evaluated by performing a cross product of the tangent vector along the s-gb
(formed at any voxel center by joining it to the next center in the s-gb) with
the unit vector perpendicular to the plane and then normalizing it to make it
a unit vector. The $S$ field is then extracted by evaluating the normal
‘action’ of $P$ on $n$ as

$S=\frac{1}{h}(Pn)\otimes n.$ (12)

We note that along an s-gb identified by the $|P|$ field, not all voxels are
necessarily “lighted up” by a non-zero value of the $S$ field.

###  4.5 Simulations and results

fig:padding shows the two domains and the features in them for which we
perform simulations of the theory presented in Sec. 4.2. More specifically,
fig:padding(a) shows the s-gb identified from the experimental data and the
$|P|$ threshold. This s-gb is defined based on orientation difference and
confidence values for reconstruction. The s-gb has prescribed eigenwall fields
processed from experimental data, as described above in Sec. 4.4.1. Elsewhere
in the domain the eigenwall field is prescribed to be zero. As this s-gb
belongs to a polycrystal assembly, for the purpose of obtaining an estimate of
the stress field and capture the effect of the network of grain boundaries, we
included a larger elastic domain with the same elastic properties as used for
the sample. Fig. 8(b) shows the network of the grain boundaries along with the
s-gb. The network of grain boundaries have non-zero interfacial dislocation
density defined as discussed in Sec. 4.3. Elsewhere in the domain the
dislocation density is zero.

We solve the stress equilibrium and elastic incompatibility equations, system
(8), with zero-traction boundary conditions. Both isotropic and anisotropic
simulations are carried out using a domain size of $150\times 150$ voxels;
each voxel is a square of edge length $1.8\ \mu m$. For either case, solutions
for the elastic distortion are unique up to constant skew tensor, and this is
assigned arbitrarily for the purpose of stress field evaluation. If the
distortion field is to be recovered, the known value of the experimental
orientation field can be specified at one point of the domain. An important
characteristic of our computational methodology, based on weak formulations of
the governing equation (8) implemented by adapted algorithms within the finite
element method, is that we do not have to calculate any derivatives of the $S$
field for the stress analysis, even though the calculations faithfully
represent the stress field of the g.disclination field $\mathit{\Pi}$.

To simulate the anisotropic case, we need the components of the tensor of
elastic moduli of each voxel with respect to the sample frame. For this we
express the lattice basis frame in terms of the sample frame for every voxel.
This linear combination between two orthogonal bases was directly obtained
from the reconstructed experimental data. Thus, if $(c_{i}),i=1,2,3$ is the
sample basis and $e^{A}_{i}$ is the lattice basis, and $A$ denotes a
particular voxel,

$\displaystyle\eqalign{&e^{(A)}_{i}=\alpha^{(A)}_{ij}c_{j}\Rightarrow
e^{(A)}_{i}\cdot c_{k}=\alpha^{(A)}_{ik};\\\
&\mathbbm{C}^{(A)}=C^{(A)}_{ijkl}e^{(A)}_{i}\otimes e^{(A)}_{j}\otimes
e^{(A)}_{k}\otimes e^{(A)}_{l}=\tilde{C}^{(A)}_{mnst}c_{m}\otimes c_{n}\otimes
c_{s}\otimes
c_{t}\Rightarrow\tilde{C}^{(A)}_{mnst}=C^{(A)}_{ijkl}\alpha^{(A)}_{im}\alpha^{(A)}_{jn}\alpha^{(A)}_{ks}\alpha^{(A)}_{lt}.\\\
}$

For both the isotropic (Fig. 11) and anisotropic cases (Fig. 12), we find, as
expected, that the stress field has large magnitude in the region where the
$S$ field varies along the interface and especially at terminations. The
stress field is almost the same in the ‘s-gb+gb’ and the ‘only-s-gb’ cases due
to the grain boundary network not producing significant stresses. Fig. 10
shows the log of the norm of the stress field produced just by the network of
grain boundaries; $G$ denotes the shear modulus. Despite the interfacial
dislocation density of the gb network being highly inhomogeneous, vanishing
stresses corresponding to it are calculated.

[scale=.70]figures/SGB_S_norm.png

Figure 9: $\log_{10}|S|$ in the domain for the isolated s-gb case. The
background (domain apart from s-gb) have $S=0$.

[scale=.70]figures/GB_iso_net_2.png

Figure 10: Result for isotropic simulation is plotted for just the network of
grain boundaries calculated using skew-symmetric $U^{e}$. Note the different
color scale used here.

[scale=.48]figures/SGB.png(-8,5)(a)
[scale=.48]figures/SGB_GB_iso_net.png(-8,5)(b)

Figure 11: Results for isotropic simulation is plotted for (a) the s-gb only
and (b) the s-gb in the presence of large angle grain boundaries. The stress
field is normalized by the shear modulus so zero on the color scales
correspond to a stress equal to the modulus.

[scale=.48]figures/SGB_Aniso.png(-8,5)(a)
[scale=.48]figures/SGB_GB_aniso_net.png(-8,5)(b)

Figure 12: Results for anisotropic simulation is plotted for (a) the s-gb only
and (b) the s-gb in the presence of large angle grain boundaries.

The non-zero stress field away from the sub-grain boundary, in Fig. 11 and
Fig. 12, is evidence of the fact that the field caused by a g.disclination
extends well beyond its core. This observation can be further extended to
infer that two g.disclinations can strongly interact even if they are
macroscopically separated and, in fact, the force of this interaction is
stronger than that of an equally separated pair of dislocations.

The stress fields for the isotropic and the anisotropic cases are similar in
magnitude and spatial distribution, and are sufficient to induce plasticity.
This observation holds for both the cases defined above. For the 2-d
approximation used here (see Sec. 4.4) the elasticity tensor relates in-plane
stress to the planar strain. For 2-d, $C_{ijkl}$, $i,j,k,l={1,2}$ has 16
coefficients. Due to major and minor symmetries in the indices, we have 6
unique coefficients in the elasticity tensor. These coefficients are required
to calculate the elastic fields. Even though the material is highly
anisotropic, for the plane of interest used here (see Sec. 4.4) these
coefficients are similar for both the cases. This yields similar elastic
fields.

##  5 Discussion

An HEDM measured dataset has been analyzed to isolate s-gb features based on
discontinuities (at the scale of observation) in the orientation field. These
features have then been used to calculate associated elastic fields. Sources
for the elastic fields arise from longitudinal variations of the rotation jump
across the boundary, specified from the analysis of the experimental data.

Limitations in calculating the exact shape of the feature arise from the
disparity in s-gb thicknesses and the resolution of the observations. Further,
noise in the measured orientation values will tend to generate computed
stresses. For example in Fig. 9, while longer length scale variations appear
systematic, there are local (nearest neighbor) variations which may or may not
be. Characterization of the results of various smoothed structures will be
evaluated in future work.

Further limitations of translating available experimental data to input for
elastic field calculations lies in the manual intervention required to
determine an s-gb manifold (in this work, e.g., the calculation of the tangent
along a best approximated s-gb curve as shown in Fig. 3). More fully automated
procedures can be developed to locate and characterize features throughout the
measured volume of microstructure.

The calculations of defect elastic fields have been restricted to 2-d, small
deformation, linear elasticity. A computational capability for 2-d, finite
deformation calculations of g.disclination defect fields is available to us
[22]. As well, finite strain, dislocation plasticity codes in 2 and 3-d have
also been developed by us [37, 38, 39]. While theory and algorithms for 3-d,
g.disclination based elastic field calculations have been developed by us
[22], a computational implementation for 3-d calculations needs to be
developed. Finally, dislocation plasticity calculations in 3-d are
computationally expensive, and more efficient strategies for parallel
computation are needed to realize the potential of our combined experimental-
theoretical advances.

##  6 Conclusion

This proof-of-principle calculation demonstrates a method that takes advantage
of spatially resolved experimental measurements of unit cell orientation
fields in three dimensions. The measurements include intra-granular
orientation variations as well as rotations that naturally occur at grain
boundaries. We calculate purely elastic stress fields of the defects that
arise due to structural heterogeneities in orientation fields, without
incorporating any plastic relaxation in this effort. Obviously, this can be
improved. But these stresses represent precursor driving forces for localized,
heterogeneous plasticity in and around sub-grain boundaries that perforce must
exist in the material, and be mitigated/relaxed, since the orientation fields
they correspond to persist for long time periods after real-time, step-wise
tensile extension of the sample.

Future experimental analysis work will attempt to include the evaluation of
inhomogeneous strain fields within grains [29]. One example of a topologically
interesting unit cell orientation discontinuity has been analyzed here. Future
modeling work will extend the calculations to three dimensions and apply the
methodology to cases where multiple orientation discontinuities occur within a
single grain. A grand and glorious goal is to use the full power of mesoscale
field dislocation mechanics [37, 38, 39] to model sample evolution from the S0
state to S2 of Fig. 1 and/or analogous evolution in other materials. Such
exercises will help shed light on the prediction capabilities of our
theoretical-computational models leading to their further refinement, as well
as raise interesting questions for extending experimental measurements to more
incremental sequences of states and to the use of complimentary techniques
with higher spatial resolution in nanoscale grains in small samples [40].

## Acknowledgments

This work was supported by the grant NSF OIA-DMR #2021019. The authors declare
that they have no known competing financial interests or personal
relationships that could appear to have influenced the work reported in this
paper.

##  Appendix A Dislocation density of an infinitesimal rotation field cannot
capture terminating interfaces

We show here that interfacial dislocation density fields calculated from skew-
symmetric elastic distortion fields as their curl cannot represent terminating
interfaces (defined as formed from a rapid variation of the elastic rotation
field over a small width, as for a gb or an s-gb).

Let the axial vector field of the skew symmetric tensor field $\omega^{(e)}$
be $\Phi$:

$\omega^{e}=X\,\Phi;\qquad(\omega^{e}_{ij}=\epsilon_{ijk}\Phi_{k}).$

We will refer to the interfacial dislocation density field here as
$\alpha^{(i)}$. It is closely related to the object $S:X$ discussed in Sec.
4.2; for an interface defined by a transition between two planar elastic
distortion fields $U^{1}$ and $U^{2}$ in the direction $n$ over a width $h$

$U_{ij,k}\approx\frac{U^{1}_{ij}-U^{2}_{ij}}{h}n_{k};\qquad\alpha^{(i)}_{il}=(\mbox{curl}U)_{il}=\epsilon_{lkj}U_{ij,k}\approx-\frac{(U^{1}-U^{2})_{ij}n_{k}}{h}\epsilon_{jkl}.$

When the elastic distortion field is skew,

$\alpha^{(i)}=\mbox{curl}(\omega^{e})=\mbox{curl}(X\Phi)=(\mbox{grad}\Phi)^{T}-(\mbox{div}\Phi)I;\qquad(\alpha^{(i)}_{im}=\epsilon_{mjk}\omega^{e}_{ik,j}=\Phi_{m,i}-\delta_{im}\Phi_{n,n}),$

where $I$ is the identity tensor. Let $\Phi=\phi e_{3}$, where $e_{3}$ is the
out-of-plane direction. Consider now an arbitrary parametrization of the plane
by the coordinates $\left(\xi^{1},\xi^{2}\right)$ with natural basis denoted
by $\left(a_{i}:=\frac{\partial x}{\partial\xi^{i}},i=1,2\right)$ and dual
basis $\left(a^{i}:=\frac{\partial\xi^{i}}{\partial x}\right)$. Then

$\displaystyle\alpha^{(i)}$ $\displaystyle=(\mbox{grad}(\phi
e_{3}))^{T}-(\mbox{div}(\phi e_{3}))\mbox{I}$
$\displaystyle=\left[\frac{\partial(\phi
e_{3})}{\partial\xi^{i}}\otimes\frac{\partial\xi^{i}}{\partial
x}\right]^{T}-\left[\mbox{tr}\left(\frac{\partial{(\phi
e_{3}})}{\partial\xi^{i}}\otimes\frac{\partial{\xi^{j}}}{\partial
x}\right)\right]\mbox{I}$ $\displaystyle=\left[\frac{\partial\xi^{i}}{\partial
x}\otimes\frac{\partial(\phi
e_{3})}{\partial\xi^{i}}\right]-\left[\frac{\partial{\phi}}{\partial\xi^{i}}e_{3}.\frac{\partial{\xi^{j}}}{\partial
x}\right]\mbox{I}$

and, since both the primal and dual basis vectors are in-plane and
perpendicular to the $e_{3}$ direction,

$\alpha^{(i)}_{i3}a^{i}\otimes
e_{3}=\alpha^{(i)}=\frac{\partial\phi}{\partial\xi^{i}}\left[a^{i}\otimes
e_{3}\right]\Longrightarrow\alpha^{(i)}_{i3}=\frac{\partial\phi}{\partial\xi^{i}},$
(14)

where $\alpha^{(i)}_{i3}$ are the non zero components of the interfacial
dislocation density tensor on the (generally spatially varying) dual basis
$a^{i}\otimes e_{3},i=1,2$.

By the equality of mixed partial derivatives, we then have

$\alpha_{13,2}=\alpha_{23,1}.$ (15)

Thus, the components, w.r.t an arbitrary coordinate system, of the interfacial
dislocation density tensor generated from a rotation field must satisfy the
relation (15).

The condition (15) holds in any coordinate system, including a rectangular
Cartesian one. Geometric intuition for the condition is obtained by defining
the vector field $\rho:=\hat{\alpha}_{13}\rme_{1}+\hat{\alpha}_{23}\rme_{2}$,
where the $\hat{\alpha}_{i3}=\alpha_{i3}$ for $(\xi^{1},\xi^{2})$ a
rectangular Cartesian coordinate system with basis $(e_{1},e_{2})$. Then

$\hat{\alpha}_{13,2}-\hat{\alpha}_{23,1}=0\Rightarrow\mbox{curl}\,\rho=0\Rightarrow\rho=\mbox{grad}\,\theta$
(16)

(we assume the region to be simply-connected) for some scalar field $\theta$,
and (14) implies that $\theta=\phi$ (up to an irrelevant constant).

This geometrically means that the vector field $\rho$ points in the normal
direction to level sets (contours) of the scalar field $\phi$, as shown in
Fig. 13.

[scale=.04]figures/Layer_1.png [scale=.45]figures/grain_2.png

Figure 13: A curved layer consisting of level sets of $\phi$ with the normal
representing the direction of $\rho$ at that point.

We now show that a layer with a specified interfacial dislocation density
generated from a skew symmetric elastic distortion field cannot terminate,
which is equivalent to the statement that if a layer interfacial dislocation
density field terminates, it is not generated from a skew symmetric elastic
distortion field (i.e. an infinitesimal rotation field).

Assume that an interfacial layer is generated from a skew-symmetric
$\omega^{e}$. This means that the field $\rho=\mbox{grad}\,\phi$ is
concentrated in the layer. But then the level contours of $\phi$ must be
parallel to the layer, as otherwise the contours of $\phi$ would have to
terminate at the lateral boundaries of the layer suggesting that
$\mbox{curl}\,\rho$ would have to be non-vanishing at these boundaries (since
there would be a gradient, in the direction parallel to the contours, of the
$\rho$ field, which is normal to the contours), and this is not possible by
(16) - see Fig. 14. But, by the same logic, since the contours of $\phi$ are
parallel to the (generally curved) interface, the interface itself cannot
terminate as again, in that case, $\mbox{curl}\,\rho\neq 0$ at such a
termination, and this is impossible. This proves the claim.

Thus, quantitatively speaking, $\mbox{curl}\,\alpha^{(i)T}$ may be considered
as an indicator of layer termination and this field must vanish when the
interfacial dislocation density field in question, $\alpha^{(i)}$, is
generated from a skew symmetric field. This is consistent with the fact that
in linear isotropic elasticity, the source of internal stress is given by the
field $\left(\mbox{curl}\,\alpha^{(i)T}\right)_{sym}$ [41].

[scale=.45]figures/Fig_appendix_arg_1.png

Figure 14: Sketch for proving $\phi$ contours have to be parallel to layer by
contradiction. $\rho$ has a component along the layer producing non-zero
$\mbox{curl}\,\rho$.

A related characterization of the result is as follows: assume that a
terminating boundary has a distribution of interfacial dislocation density on
it in the general class indicated below in (A). Then it cannot satisfy the
condition (15), and hence it cannot originate from a skew-symmetric elastic
distortion field. This is equivalent to the assertion that if an interfacial
dislocation density originates from a skew-symmetric $U^{e}$ field then it
cannot belong to the class (A). The argument is provided below.

Let $\tilde{\alpha}=\tilde{\alpha}_{13}\,a^{1}\otimes
e_{3}+\tilde{\alpha}_{23}\,a^{2}\otimes e_{3}$ be a specified dislocation
density on a terminating layer parametrized by curvilinear coordinates
($\xi^{1},\xi^{2}$) such that $\xi^{2}$ is constant along the layer, and the
components of $\tilde{\alpha}$ on the dual basis $a^{i}$ are given by

$\displaystyle\tilde{\alpha}_{13}(\xi^{1},\xi^{2})$
$\displaystyle=A_{1}H(\xi^{1})[H(\xi^{2}-l/2)-H(\xi^{2}+l/2)]$
$\displaystyle\tilde{\alpha}_{23}(\xi^{1},\xi^{2})$
$\displaystyle=A_{2}H(\xi^{1})[H(\xi^{2}-l/2)-H(\xi^{2}+l/2)],$ (17)

where $l$ is the thickness of the layer and $A_{1}$ and $A_{2}$ are constants,
see Fig. 15. $H(x-x_{0})$ represent a Heaviside step function centered at
$x_{0}$. Then,

$\displaystyle\tilde{\alpha}_{13,2}(\xi^{1},\xi^{2})$
$\displaystyle=A_{1}H(\xi^{1})[\delta(\xi^{2}-l/2)-\delta(\xi^{2}+l/2)]$
$\displaystyle\tilde{\alpha}_{23,1}(\xi^{1},\xi^{2})$
$\displaystyle=A_{2}\delta(\xi^{1})[H(\xi^{2}-l/2)-H(\xi^{2}+l/2)]$

where $\delta(x-x_{0})$ represents a Dirac delta function centered at $x_{0}$,
and it is clear that $\tilde{\alpha}_{23,1}$ cannot equal
$\tilde{\alpha}_{13,2}$, irrespective of the values for the constants $A_{1}$
and $A_{2}$ (except when both are zero) (this argument does not change in
substance if $A_{i}$ are assumed to be functions of $\xi^{1}$).

[scale=.35]figures/Fig_appendix_2.png [scale=.48]figures/Mag_app_2

Figure 15: Parametrization of a terminating layer by $\xi^{1}$ and $\xi^{2}$,
and a zoomed in version at the termination.

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the relationship between scale factor and time is that the EDE universe
reaches a fixed scale factor at an earlier time because of the component of
accelerated expansion — note that the ratio $t_{\rm EDE}/t_{\rm Pl}$ starts at
unity in the period before EDE becomes dynamically significant. At $a_{\star}$
— which, crucially, does not change between the two models for the relatively
low values of $f_{\rm EDE}$ being considered — the EDE universe is 4.7%
younger than the Planck universe. A sound wave with a given velocity $c_{\rm
s}(t)$ will therefore not travel as far in EDE as in Planck, reducing
$r_{\star}$ in EDE by 4.1% relative to Planck. (The sound speed differs very
slightly between the two models, but this is only a 0.3% effect at
$a_{\star}$; this difference declines at smaller scale factors and is
therefore always subdominant to the change in $a(t)$ between the two models.)
While the physical sound horizon is slightly larger at a given time in EDE
than in Planck — as one might expect for a Universe that is has a component
that is acting as dark energy — this is overwhelmed by the change in $a(t)$,
meaning the comoving sound horizon is larger in Planck at fixed time. The
change in $t(a_{\star})$ between the two models further increases the
difference in the comoving sound horizon.