# A taxonomy of grain boundary migration mechanisms via displacement texture
characterization

Ian Chesser<EMAIL_ADDRESS>Brandon Runnels<EMAIL_ADDRESS>Elizabeth
Holm<EMAIL_ADDRESS>Department of Materials Science and Engineering,
Carnegie Mellon University, Pittsburgh, PA, United States  Department of
Mechanical and Aerospace Engineering, University of Colorado, Colorado
Springs, CO, United States  Department of Physics and Astronomy, George Mason
University, Fairfax, VA, United States

###### Abstract

Atomistic simulations provide the most detailed picture of grain boundary (GB)
migration currently available. Nevertheless, extracting unit mechanisms from
atomistic simulation data is difficult because of the zoo of competing,
geometrically complex 3D atomic rearrangement processes. In this work, we
introduce the displacement texture characterization framework for analyzing
atomic rearrangement events during GB migration, combining ideas from slip
vector analysis, bicrystallography and optimal transportation. Two types of
decompositions of displacement data are described: the shear-shuffle and min-
shuffle decomposition. The former is used to extract shuffling patterns from
shear coupled migration trajectories and the latter is used to analyze
temperature dependent shuffling mechanisms. As an application of the
displacement texture framework, we characterize the GB geometry dependence of
shuffling mechanisms for a crystallographically diverse set of mobile GBs in
FCC Ni bicrystals. Two scientific contributions from this analysis include 1)
an explanation of the boundary plane dependence of shuffling patterns via
metastable GB geometry and 2) a taxonomy of multimodal constrained GB
migration mechanisms which may include multiple competing shuffling patterns,
period doubling effects, distinct sliding and shear coupling events, and GB
self diffusion.

##  1 Introduction

Grain boundaries (GBs) are mobile planar defects in polycrystalline materials
with structures and motion mechanisms that comprise a large phase space.
Controlling GB structure and mobility is an important challenge during the
processing of metallic and ceramic materials [1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12]. Significant progress has been made in GB migration science over the
last several decades at the atomic scale via a combination of atomistic
simulations and experiments. Atomistic simulations, augmented by increasingly
powerful computing resources and data analysis tools, have enabled high-
throughput surveys of grain boundary properties and the discovery of new grain
boundary structures and migration mechanisms [13, 14, 15]. In-situ imaging of
grain boundary migration at the atomic scale has confirmed several theories
that, until recently, were only supported by molecular dynamics (MD)
simulations. For example, simultaneous grain rotation and shrinkage was
recently observed in a 2-D system, confirming a main prediction of the Cahn-
Taylor model [16, 17]. Disconnection nucleation and growth has been observed
directly during shear driven motion of grain boundaries in bicrystals [18].
Despite these synergistic findings, experiments raise hard questions about the
impact of constraints and coupled processes on GB migration in real
microstructures. Low dimensional structural phase transformations at GBs can
dramatically impact grain boundary mobility [15, 19, 20]. Large scale grain
growth experiments show that the same crystallographically defined GB can move
at very different speeds depending on local microstructure [21, 22]. The
impact of metastable GB structure and constraints on GB migration is still not
well understood at the atomic scale.

There have been a variety of post-processing techniques applied to MD
simulations to characterize atomic displacement patterns in space and time
during GB migration. Spatial displacement analysis methods include slip vector
analysis [23, 24, 25], microrotation analysis and other computations inspired
from continuum mechanics [26, 27]. These methods have been particularly useful
in analyzing ordered shuffling patterns of low $\Sigma$ GBs, where it has been
concluded that ordered shuffling patterns are a necessary but insufficient
condition for fast, non-thermally activated grain boundary migration [25].
Such ordered atomic rearrangement processes resemble those occurring during
twinning and martensitic phase transformations [28, 29]. Although these
spatial displacement analysis methods are agnostic to unit mechanism, they are
difficult to interpret in noisy settings where multiple mechanisms may compete
at high temperatures or stresses.

Another class of spatiotemporal methods from statistical physics enables
analysis of characteristic length and time scales associated with GB
migration. The Van Hove correlation function and self intermediate scattering
function, when computed over time intervals spanning atomic vibrations and
atomic diffusion, are useful measures of dynamic heterogeneity and structural
relaxation occurring during GB migration [30, 31, 32, 33]. On the basis of
these techniques, it has been argued previously that atomic rearrangements
during GB migration bear strong resemblance to dynamic heterogeneity in
supercooled glass forming liquids [30, 31]. For instance, quasi-1D clusters of
mobile atoms called “strings” were observed during migration that increase in
average length with decreasing temperature, similar to supercooled liquids
[32]. Stringlike displacements have also been observed during GB self
diffusion [34]. Although it is is well known that GB diffusion and migration
can coexist, it is unclear how different types of atomic rearrangements such
as strings and collective shuffles limit the rate of boundary migration under
different conditions including varying interface geometry, temperature and
driving force. Although spatiotemporal analysis methods are very useful for
analyzing GB migration, they require migration trajectories with high time
resolution that are expensive to analyze for a large parameter space.

In this work, we aim to infer grain boundary migration mechanisms from two
snapshots of grain boundary motion before and after motion. This set-up is
similar to comparing TEM images before and after migration, except that in MD
simulations the identities of distinct atoms are known. We approximate GB
migration trajectories as probability distributions over shuffling vector
lengths and orientations in the dichromatic pattern. We call these
distributions displacement textures, a term inspired by spin textures in
magnetic materials. The displacement texture characterization procedure for
analyzing atomic shuffles during GB migration can be seen as a 3-D
generalization of slip vector analysis. Unlike standard slip vector analysis,
however, displacement texture characterization accounts for multiple reference
frames associated with symmetries in the dichromatic pattern, similar to the
topological model of Hirth and Pond [28, 35]. Our framework allows inference
of shuffling patterns for distinct disconnection modes. Disconnections are
unit defects responsible for shear coupled migration and possess both sliding
(burgers vector) and rotational (step) character. Multiple disconnection types
with closely spaced nucleation and migration energy barriers may compete at
finite temperature, leading to interesting phenomena such as GB roughening (a
structural transition which leads to the onset of significant motion at finite
temperature) and zig-zag mode switching under constraints [36]. It is shown in
this work that displacement texture analysis allows inference of distinct
disconnection types from multi-modal MD trajectory data where traditional
indicators of disconnection type such as coupling factor are difficult to
interpret.

In order to construct a taxonomy of GB migration mechanisms via displacement
texture analysis, we consider a diverse set of migration scenarios including
the motion of twin boundaries, disconnection mediated shear coupled motion,
and constrained mixed mechanism scenarios involving shuffling, sliding and GB
self diffusion. In combination with displacement texture characterization, a
main analysis method of this work is a recently introduced forward model for
GB migration which enables the interpretation of noisy and multimodal
displacement data [37]. The main idea of the forward model is to generate a
large catalog of GB migration mechanisms to compare to MD data. The forward
model is informed by bicrystallography and allows enumeration of spatial
displacement patterns for different misorientations, microscopic translations
and disconnection modes. The forward model is probabilistic, enabling the
prediction of competing transformations in the dichromatic pattern with a
heuristic temperature-like regularization parameter. It is shown in this work
that the forward model greatly aids the interpretation of displacement
textures from MD data.

The displacement texture method is formally described in the first half of
this paper with examples given in the second half. Two types of decompositions
of displacement data are described in Section 3: the shear-shuffle and min-
shuffle decomposition. The former is used to extract shuffling patterns from
shear coupled migration trajectories and the latter is used to analyze
temperature dependent shuffling mechanisms. As an application of the
displacement texture framework, we characterize the GB geometry dependence of
shuffling mechanisms for a crystallographically diverse set of mobile GBs in
FCC Ni bicrystals. Four examples of scientific interest include analysis of
the boundary plane dependence of shuffling for $\Sigma 3$ GBs, analysis of
variation of shuffling patterns with misorientation, analysis of mixed sliding
and migration in a $\Sigma 11$ twist GB, and analysis of roughened, multimodal
migration of symmetric tilt GBs.

##  2 Molecular dynamics methodology

We use the FCC Ni bicrystal dataset generated by Olmsted et. al in [13, 38] as
a starting point for MD simulations. To illustrate how migration mechanisms
vary with crystal structure, we also use several bicrystal structures from the
BCC Fe dataset generated by Ratanaphan et. al [39]. These datasets span a
variety of misorientations and boundary plane inclinations and are used to
explore the impact of GB geometry on migration mechanisms. Boundary conditions
for bicrystal structures were chosen to be periodic in the plane of the GB and
free normal to the GB plane with 1 nm thick rigid slabs defined at each free
surface. All bicrystals were resized to have dimensions larger than 16 nm in
the interface plane normal direction ($\hat{x}$ direction) with cross sections
larger than $3$ nm $\times$ $3$ nm. All GB structures were previously
optimized via a grid search over microscopic translations $\bm{p}$ such that
free energy is minimized at 0 K [13]. FCC Ni structures employ the Foiles-Hoyt
EAM Ni potential [40] as a model for interatomic bonding. The melting point of
this potential is 1565 K. BCC Fe structures employ the Fe EAM Mendeleev
potential [41]. The melting point of this potential is 1772 K. Molecular
dynamics simulations were performed in LAMMPS [42] and visualization performed
in OVITO [43].

The ramped synthetic driving force method (rSDF) [44] is used to generate
grain boundary motion in bicrystal structures under both free and fixed
boundary conditions. When fixed boundary conditions are enforced, a slab at
the right end of the bicrystal is held in place. Before application of a
driving force, relaxed bicrystal structures are equilibrated in a multistep
procedure at a given set temperature. First, 0 K bicrystal structures are
uniformly expanded by a pre-computed thermal expansion coefficient. Next, they
are equilibrated in the NVT ensemble with free boundary conditions for 0.1 ns.
The slab at the right end of the bicrystal is allowed to float in the plane
normal direction to relieve stress in that direction. At the end of this
procedure, normal stresses in the bulk grains are verified to fluctuate around
zero with small amplitudes. After the equilibration run, boundary conditions
are imposed (free or fixed) and a synthetic driving force is applied as a
potential energy jump across the GB. The synthetic driving force is ramped
upward from 0 at a rate of 0.2 meV/ps [45] with the energy jump applied
symmetrically across the GB. Each simulation run lasted 0.2 ns or was ended
early if the GB had moved more than 4 nm. rSDF simulations for FCC Ni GBs were
performed in the low to intermediate temperature range 100-1000 K (0.06-0.65
$T_{m}$). This temperature range was selected for ease of analysis of unit
mechanisms.

Depending on GB geometry, critical driving force values were found to vary
from experimentally accessible driving force magnitudes (less than 10 MPa) for
highly mobile GBs to high values (on the order of GPa) for low mobility GBs
[46, 47, 45]. Critical driving force values generally decrease with
temperature, as described in [47]. Values of critical driving forces for free
boundary conditions were previously tabulated in [45]. The main goal of this
work is analysis of migration mechanisms.

The ramped driving force is useful for stimulating changes in migration
mechanism during a single simulation run. In particular, GBs that would
ordinarily stagnate under constrained boundary conditions at constant driving
force are often observed to continue migrating as driving force is increased
via a diverse array of mode switching mechanisms. Our taxonomy of mode
switching behavior extends the observations of zig-zag disconnection mode
switching by Spencer and Srolovitz to a larger class of GBs [36].

The Janssens implementation [48] of the synthetic driving force was used in
this work because of its computational efficiency. An individual 0.2 ns run of
the rSDF method took several hours on a 64 cpu machine compared to the
original ECO method [49] which took several days per run. The Janssens driving
force has been shown to give consistent migration mechanism results compared
to other techniques, including strain driven motion [23], random walk motion
[50] and the ECO implementation of the synthetic driving force [49].
Nevertheless, a deficiency of the Janssens implementation is that nearest
neighbor rearrangement events of atoms in the GB core can lead to
discontinuous changes in the applied synthetic energy which violate
conservation of energy in the simulation. Mobilities computed with the ECO and
Janssens method have shown similar trends with temperature and driving force
but have systematic differences in magnitude with Janssens mobility values
being lower than ECO values. At the time of writing, a more computationally
efficient version of the ECO driving force has been published with
significantly faster force calculations [51] compared to the original version.
Consistency of the mechanism results from this work with the new ECO method
should be checked in future work.

##  3 Displacement Texture

###  3.1 Informal statement of main ideas

Colored displacement textures highlight strikingly ordered patterns of atomic
motion during grain boundary migration (Figure 1). Displacement textures
comprise the linear net displacement vectors obtained by subtracting the
atomic positions after GB migration from those before GB migration. The goal
of displacement texture analysis in this work is twofold: 1) infer and
separate migration mechanisms within individual GB migration trajectories and
2) quantify how displacement patterns vary over a large parameter space
including different GB types, temperatures, driving forces and variable
boundary conditions.

The displacement textures in Figure 1 illustrate a diverse range of migration
mechanisms. Some GBs, such as the $\Sigma 45$ $\hkl<210>$ $84^{\circ}$
symmetric tilt GB in BCC Fe (top left), exhibit visually striking helical
displacement patterns that tile space in 3D. The visually apparent order of
these displacement patterns stands in stark contrast to the disordered core
structure of the GB. Other GBs, such as faceted $\Sigma 3$ mixed tilt-twist
GBs (lower left), exhibit ordered layers of displacements. Many tilt and twist
GBs sweep out vortex displacement patterns which circulate around fixed points
of near zero displacement (upper and lower right). In some cases, such as
migration of the $\Sigma 5$ $\hkl(430)$ asymmetric tilt (lower middle) and
$\Sigma 51$ (110) 16∘ twist GB (lower right), networks of displacements are
apparent that are distinct from the overall ordered displacement pattern.
These mixed displacement types indicate the coexistence of GB migration and GB
self diffusion. Nonuniform displacement textures are indicative of a change in
migration mechanism or a mixture of migration mechanisms and are common during
migration simulations with variable driving force [45]. An example of
nonuniform migration is the constrained, driven migration of a $\Sigma 37$
¡100¿ 18.9∘ symmetric tilt GB (top right) which shows shear deformation during
an early part of its motion and vortex displacement patterns with less
apparent shear later in its motion. Interpretation of non-uniform migration
trajectories is a characterization challenge which requires defining and
separating unit mechanisms.

Figure 1: These images show diverse displacement textures swept out by mobile
GBs considered in this work. Atoms in the grain boundary core are shown
explicitly, colored by non-FCC coordination (HCP = light blue, BCC = grey, SC
= white, OTHER = dark blue) in OVITO [52]. Atomic displacements are colored by
orientation in the sample frame of each bicrystal using the default ipfHSV
color scheme in MTEX [53]. The scale of the top left image is 5 nm x 3 nm. Top
left: $\Sigma 45$ ¡210¿ 84∘ tilt, BCC Fe (id 102), 100 K. Top right: $\Sigma
37$ ¡100¿ 18.9∘ tilt, FCC Ni (id 216), 200 K. Bottom left: $\Sigma 3$ mixed
tilt twist, FCC Ni (id 337), 500 K. Bottom middle: $\Sigma 5$ (430) asymmetric
tilt, BCC Fe (id 36), 500 K. Bottom right: $\Sigma 51$ (110) 16∘ twist GB, FCC
Ni (id 356), 400 K. Boundary conditions are fixed in all of the simulation
snapshots shown above. For full crystallographic information corresponding to
each GB id, see the supplementary information of [13, 39]

To allow for quantitative analysis of displacement patterns during GB
migration, we will formally define displacement texture as a probability
distribution over shuffling lengths and directions in the dichromatic pattern
in Section 3.5. In Sections 3.2 and 3.3, we give prerequisite background
information about the dichromatic pattern and the shear-shuffle decomposition.
The shear-shuffle decomposition allows identification of shuffle displacements
corresponding to arbitrary disconnection modes. In Section 3.4, we review key
aspects of a previously developed forward model for shuffle patterns [37] to
set the stage for interpreting the impact of geometric degrees of freedom
(microscopic shift, misorientation and boundary plane inclination) on
migration mechanisms. Details of practical implementation of displacement
texture analysis are given in 3.6.

The main graphical representation of displacement texture in this work is an
inverse pole figure which we call the displacement pole figure (Figure 2). In
the displacement pole figure, orientations of displacement vectors swept out
during GB migration are plotted in stereographic projection in the sample
frame of the bicrystal. Further details of the practical implementation of
displacement texture analysis are given in 3.6.

Figure 2: (a) Example of a displacement texture for a $\Sigma 7$ GB (id 26) at
300 K under constrained boundary conditions. The hexagonal shuffling pattern
highlights the (111) symmetry of the dichromatic pattern for this GB. Atoms at
fixed points at the center of each hexagon do not need to move to accommodate
reorientation and correspond to coincident cites in the dichromatic pattern
(b) The displacement texture is represented by stereographic projections of
the displacements in the upper and lower hemisphere (upper panel). In this
case, the six non-zero shuffling vectors are equally probable but have
different displacement lengths (lower panel), all of which are shorter than
the nearest neighbor distance $\sigma=2.49$ angstroms. Error bars are computed
as the standard deviation in displacement length for the set of atoms with a
given shuffle orientation. Asymmetric shuffle lengths result from a preferred
microscopic translation in the dichromatic pattern.

###  3.2 The coherent and translated dichromatic pattern

Atomic displacements swept out during conservative grain boundary migration
connect sub-lattices of the dichromatic pattern. In this section, we formally
define two types of dichromatic patterns useful for analyzing GB migration,
borrowing terminology from the topological model of Hirth and Pond [54, 55,
56].

Let $\mathcal{X},\mathcal{Y}\subset\mathbb{R}^{3}$ be the set of all atomic
positions in the lattice 1 and lattice 2 configurations of a bicrystal. We
choose the reference atomic positions $\mathcal{X},\mathcal{Y}$ such that
$|(\mathcal{X}\cap\mathcal{Y})|\geq 1$; in other words, there is at least one
common (“coincident”) site (Equality holds in the case of lattices rotated by
an irrational angle, since the only coincident point is at the origin [57]).
$\mathcal{X}\cap\mathcal{Y}$ is the Coincident-Site Lattice (CSL) for the
bicrystal and the union $\mathcal{X}\cup\mathcal{Y}$ is the Coherent
Dichromatic Pattern (CDP). The CDP is the most commonly depicted dichromatic
pattern in the literature, but is not the reference frame that is most
relevant for grain boundary migration.

In real or simulated boundaries, there is also a rigid microscopic shift
($\bm{p}\in\mathbb{R}^{3}$) between the two sub-lattices in the dichromatic
pattern, which is conventionally understood to minimize the grain boundary
energy for a stationary boundary. The actual atomic positions in lattice 2 are
denoted $x=\mathcal{Y}+\bm{p}$. 111The use of lowercase $x$ is following
traditional continuum mechanics parlance that denotes undeformed and deformed
positions by uppercase and lowercase letters, respectively. It is useful to
construct a modification of the CDP: $\mathcal{X}\cup x$, which is referred to
as the Translated Dichromatic Pattern (TDP). The raw atomic displacements
swept out by GB migration during MD simulations are represented in the TDP.
Displacements in the TDP may be converted to the CDP by subtraction of the
microscopic shift vector $\bm{p}$. The CDP provides a convenient reference
frame for analyzing the crystallography of shuffling directions (see results
for $\Sigma 3$ GBs in Section 4.1.1). However, since $\bm{p}$ is often unknown
in practice (for GBs in a polycrystal), displacements in the TDP are most
directly accessible to simulations and experiments.

Figure 3: (a) The shear shuffle decomposition is shown for three different
types of transformations in the dichromatic pattern for the same boundary: a
positive coupling mode ($\beta>0$), a zero shear transformation ($\beta=0$)
and a negative coupling mode ($\beta<0$). The net deformation $\bm{F^{GB}}$
may be decomposed into a combination of shuffle displacements (middle frame)
and shear displacements (arrows). The shuffling dichromatic pattern (SDP) is
associated with the zero shear transition state for a given deformation. (b)
provides a summary of dichromatic pattern types for a fixed choice of shear
deformation $\bm{F^{GB}}$ and normal coordinate $\bm{u_{0}}$. These three
types of dichromatic patterns are distinguished by three rigid translations in
the SDP. The coherent dichromatic pattern (CDP) is associated with zero
translation. The translated dichromatic pattern (TDP) is obtained from the CDP
by a microscopic shift $\bm{p^{*}_{GB}}$ typically associated with energy
minimization. The optimal translated dichromatic pattern (OTDP) is associated
with zero net displacement per atom and minimal net shuffling distance and can
be obtained from the TDP by subtracting a translation $\langle\bm{u}\rangle$
or from the CDP by adding a translation $\bm{p^{*}_{M}}$.

###  3.3 The shear shuffle decomposition

The notion of a shear shuffle decomposition goes back to early work on
twinning by Bilby and Crocker [58, 59, 60]. It was recognized that the atomic
positions of two grains across a twin interface cannot always be matched by a
pure shear operation. In other words, depending on the twin orientation
relationship, shuffles are required in addition to shear displacements to
accommodate crystal rotation during twin formation (Figure 3a). In cases where
normal motion is coupled to sliding during GB migration, the raw displacements
swept out behind the moving boundary can be decomposed into shuffle and shear
displacements arising from the normal (step) and sliding (burgers vector)
character of disconnection motion. In this section, we describe the process of
identifying shear and shuffle displacements directly from MD data for
arbitrary disconnection types.

Given a grain boundary shear deformation gradient $\bm{F}^{GB}\in GL(3)$
(swept by reorientation of grain 1 to grain 2), general atomic displacements
$\bm{x}_{i}$ are represented in the TDP as:

$\displaystyle\bm{x}_{i}=\bm{F}^{GB}\bm{X}_{i}+\bm{p}+\bm{u}_{i}\ \ \
\bm{x}_{i},\bm{X}_{i}\in x\times\mathcal{X},$ (1)

where $\bm{u}_{n}$ is a local atomic shuffle and $\bm{p}$ is the microscopic
shift vector chosen on the basis of minimum GB energy. The grain boundary
shear deformation gradient $\bm{F}^{GB}$ usually has a single nontrivial
component in the sample frame, the scalar coupling factor
($\beta_{n}=\frac{b_{n}}{h_{n}}$), which depends on disconnection geometry
$(b_{n},h_{n})$ for mode $n$.

Atomic displacements are said to be Cauchy Borne (C-B) if the shuffling vector
$\bm{u}_{i}=\bm{0}$ and all atomic displacements point in the direction of the
net shear displacement. In the parlance of the shear shuffle decomposition,
C-B displacements are shears. Apart from a few special cases, such as shear
coupled motion of the coherent twin and other tilt boundaries [61], GB
migration (and disconnection motion) are not C-B but instead consist of both
shears and shuffles. In fact, many GBs in this survey, including most $\Sigma
3-11$ GBs, move with zero net shear deformation ($\bm{F}^{GB}=\bm{I}$) such
that shuffles dominate the reorientation geometry. It should be cautioned that
shears and shuffles are defined relative to the operative disconnection mode
or sliding mechanism. The same spatially defined atomic displacements may
participate in different disconnection modes and may therefore be classified
as shears or shuffles depending on context. It is also possible for a sequence
of alternating shear events to average to zero net deformation, as has been
observed during the zig-zag GB migration of a ¡111¿ tilt GB [36, 62, 63, 64].

Figure 4: Example of the shear shuffle decomposition for coupled migration of
the $\Sigma 25$ ¡100¿ tilt $16.3^{\circ}$ GB with an assumed (100) type
coupling mode. Raw MD data from an energy jump driven migration simulation at
100 K is shown on the left with lateral fiducial markers shown in blue-green
and atoms swept out by the moving GB deleted. In the decomposed data,
displacement vectors are colored by position into the page along the tilt axis
[65, 46].

The Shuffling Dichromatic Pattern (SDP) is the appropriate dichromatic pattern
representation for analyzing shuffling patterns from MD data. This
representation was recently introduced by Hirth in the context of twin
formation in minerals [35]. Construction of the SDP requires a shear
deformation $\bm{F}^{GB}$ related to a known coupling factor to be measured
and subtracted from the displacement pattern in the TDP. The SDP
representation is not unique, and a set of SDP representations can be indexed
by disconnection mode or coupling factor. Note that, since displacement
texture is defined as a distribution of shuffling vectors, it is necessary to
remove the affine shear deformation from the raw displacement field obtained
in MD simulations.

There are several choices for how to subtract shear from the raw displacement
pattern in order to recover shuffling vectors, including subtracting shear
from one or both sublattices in the TDP. In Figure 3a, the SDP is constructed
by symmetrically subtracting shear deformations with components $\beta_{i}/2$
and $-\beta_{i}/2$ from each of the sublattices of the TDP. In Figure 4, the
shear shuffle decomposition is applied to MD data for a $\hkl<100>$
$16.3^{\circ}$ tilt GB which exhibits shear coupling with coupling factor
$\beta=0.28$. This coupling factor corresponds to the well known (100) type
coupling mode [65, 46]. The raw displacements swept out by shear coupled
motion form a vortex pattern with increasing shear displacements away from the
vortex centers. To recover the shuffling pattern associated with this coupling
mode, shear is symmetrically subtracted about the vortex centers at a position
$\bm{u_{0}}$ normal to the GB. This subtraction recovers a layered shuffling
pattern (middle panel of Figure 4) associated with migration of this GB in the
(100) coupling mode for a specific microscopic shift $\bm{p}$. We note that
$\bm{u_{0}}$ defines a plane about which no deformation is subtracted and that
disconnection step height $h$ can be inferred from the periodicity of the SDP
with respect to $u_{0}$ [37]. A systematic numerical algorithm for calculating
$\bm{F}^{GB}$ and $u_{0}$ from bicrystal coordinates before and after
migration is described in A.

###  3.4 Optimal transportation and the min-shuffle decomposition

In this section, we describe a decomposition of shuffling vector data (in the
SDP) which has practical and theoretical significance for understanding the
impact of GB geometry on shuffling patterns. It has been hypothesized by Hirth
et. al [28] that the most likely shuffling pattern during GB migration is the
one that minimizes total shuffle distance in the dichromatic pattern. This
hypothesis was used as a starting point to develop a forward model for GB
migration based on optimal transportation in the dichromatic pattern [37]. The
mathematical framework for this forward model is described in [37]. An
important concept in this modeling framework is that shuffling patterns can be
locally optimized over microscopic shifts in the dichromatic pattern to
minimize total shuffling distance. We denote the min-shuffle transformation as
the shuffling pattern with minimum total shuffling distance for a given
disconnection mode. There exists a microscopic shift $\bm{p}=\bm{p^{*}_{M}}$
associated with the min-shuffle transformation which defines the Optimal
Translated Dichromatic Pattern (OTDP). A summary of dichromatic pattern types
is given in Figure 3b along with example shuffling patterns for a $\Sigma 3$
GB in Figure 5. The shuffling pattern in the OTDP has minimum net shuffle
distance and can easily be obtained from shuffling vectors in the TDP by
subtraction of the vector $\langle\bm{u}\rangle$ computed as the mean
shuffling vector in the TDP. $\langle\bm{u}\rangle$ can be thought of as a
sliding vector which transforms a metastable GB structure to another (possibly
unstable) structure which permits shorter shuffles.

Two important aspects of the min-shuffle transformation which will be
discussed further in this work are degeneracy and regularization. In the
degenerate case, multiple choices of $\bm{p}$ may correspond to the same
locally minimum shuffling cost in the dichromatic pattern. In this situation,
the min-shuffle transformation is not unique and different symmetric variants
of shuffling patterns may exist. One physical manifestation of this degeneracy
is given in Section 3.6 where period doubling effects are observed in the
shuffling patterns of the $\Sigma 3$(112) GB. Related to degeneracy is the
concept of regularization. The addition of regularization to the forward model
mixes multiple distinct shuffling patterns with similar shuffling costs,
allowing prediction of competing shuffling patterns in the dichromatic
pattern. The regularization parameter $\epsilon$ is a heuristic temperature-
like parameter which uniformly mixes all shuffling types in the limit
$\epsilon\rightarrow\infty$. In [37], competing shuffling patterns for the
$\Sigma 5$ (100) twist GB were observed in MD simulations which have the same
shuffle types as predicted by the regularized optimal transport model.

Figure 5: Min-shuffle transformation prediction for the $\Sigma 3$ (110) GB
with zero coupling factor $\bm{F}^{GB}=\bm{I}$. Shuffles are determined via
the optimal transport based forward model at low $\epsilon$ in the TDP with a
microscopic shift vector $\bm{p}=\bm{p^{*}_{GB}}$chosen to match the energy
minimizing shift in the relaxed bicrystal structure at 0 K. Shuffling patterns
in the CDP and OTDP are obtained by distinct rigid translations
$\bm{p^{*}_{GB}}$ and $\langle\bm{u}\rangle$ of one sublattice in the TDP. The
OTDP is distinguished by zero net displacement and shuffles with minimal total
length for this 3-shuffle transformation. The convex hull of shuffle
orientations in the OTDP is an equilateral triangle.

###  3.5 Formal definition of displacement texture

Figure 6:  Example 2D displacement textures in the SDP depicting shuffles from
(a) raw MD data (TDP) and (b) distance minimizing shuffles in the OTDP. Black
circles correspond to atoms with no displacement; blue circles are atoms prior
to displacement and red are atoms after displacement. The different colored
arrows on the left illustrate the different fundamental shuffle directions.
The right side of the figure superimposes all shuffles relocated to the
origin. The displacement texture is plotted on the black circle as a polar
plot. The dotted line corresponds to the mollified/smoothed texture
corresponding to thermal fluctuations. Although (a) and (b) both correspond to
the same type of shuffling pattern, the texture as plotted in (a) is
artificially complex, whereas (b) distills the data into its fundamental
peaks. Physically, (a) and (b) correspond to zero shear motion of the same GB
with different choices of microscopic shift.

A displacement texture is a probability distribution defined over a sphere,
such that its magnitude at each point corresponds to the average amount of
shuffle by atoms in that direction. Displacement textures enable the reduction
of large quantities of atomistic data onto a compact domain, making it
possible to identify statistically prevalent trends in atomic motion.
Displacement textures can either be calculated from atomistic simulations or
predicted using other models, and can account for discrete numbers of atoms or
continuous distribution of atomic shuffles. Figure 6 illustrates the concept
of displacement textures for an artificial 2-D shuffling pattern.

In the remainder of this section, we present a formal definition of
displacement texture in the language of measures. (While this is necessary in
order to generalize to both the discrete and continuous cases, the reader may
skip the remainder of Section 3.5 without missing anything essential to the
basic understanding of displacement textures.) We begin with the formal
definition of displacement texture, which has the structure of a measure,
defined as the following:

######  Definition 1.

The displacement texture measure $\mu$ is a measure on $S(2)$ for a set of
shuffle displacements $\\{\bm{u}_{i}\\}$ such that

$\displaystyle\int_{\varsigma\subset
S(2)}\bm{n}\,d\mu(\bm{n})=\langle\bm{u}\rangle_{\varsigma},$ (2)

where $\langle\bm{u}\rangle_{\varsigma}$ is the average atomic displacement
within $\varsigma$ and $\bm{n}$ is a unit normal vector.

For instance, if $\varsigma=\\{\bm{x}\in S_{2}:x_{1}\geq 0\\}$ (the positive
$x_{1}$ hemisphere), the action of $\mu$ on $\varsigma$ produces the average
atomic displacement in the $x_{1}$ direction. In the specific case in which a
discrete collection of shuffle displacements $\\{\bm{u}_{i}\\}$ is considered
one may define $\mu$ as a sum of Dirac measures:

$\displaystyle\mu(\bm{n})=\frac{1}{N}\sum_{i}^{N}|\bm{u}_{i}|\,\delta\Big{(}\bm{n}-\frac{\bm{u}_{i}}{|\bm{u}_{i}|}\Big{)},\
\ \ N=\\#(\\{\bm{u}_{i}\\}).$ (3)

We now define the net shuffling vector $\langle\bm{u}\rangle$,

$\displaystyle\langle\bm{u}\rangle=\int_{B\subset S(2)}\bm{n}\,d\mu(\bm{n}).$
(4)

In the special case where a finite number of discrete atoms are considered, it
is possible to show that the above formula reduces to

$\langle\bm{u}\rangle=\frac{1}{N}\sum_{i=1}^{N}\bm{u}_{i},$ (5)

recovering simply the arithmetic mean of the shuffle displacements.

Recall that shuffle vectors in the OTDP are defined as

$\displaystyle\bm{u^{*}}=\bm{u}-\langle\bm{u}\rangle$ (6)

Shuffles in the OTDP satisfy $\langle\bm{u^{*}}\rangle=\bm{0}$. In practice,
this constraint is convenient for symmetrizing displacement patterns and
comparing them across a large parameter space irrespective of small
perturbations. The shuffling vectors in the OTDP define a convex hull around
the origin:

$\displaystyle\langle\bm{u^{*}}\rangle=\bm{0}\in\operatorname{conv}\\{\bm{u^{*}}_{1},\ldots,\bm{u^{*}}_{N}\\}.$
(7)

This property is convenient for isolating and counting distinct shuffling
vectors.

The point measure definition of $\mu$ may be regularized using a mollifier
(approximated by a truncated gaussian) $\psi_{\varepsilon}$ with variance
$\varepsilon$, so that

$\displaystyle(\mu\star\psi_{\varepsilon})(\bm{n})=t(\bm{n})\mathcal{H}^{2},$
(8)

with $\mathcal{H}^{2}$ the 2-Hausdorff measure, and

$\displaystyle
t(\bm{n})=\frac{1}{N}\sum_{i}^{N}|\bm{u}_{i}|\,\psi_{\varepsilon}\Big{(}\bm{n}-\frac{\bm{u}_{i}}{|\bm{u}_{i}|}\Big{)},$
(9)

where $t(\bm{n})$ is refered to here and subsequently as the thermalized
displacement texture. Thermalization can thus be applied synthetically by
defining $\psi_{\varepsilon}$ as $\varepsilon\propto k_{B}T$. This mimics the
effect of thermal fluctuations through smearing of the displacement
poles.222We note that, since displacement texture is defined over a 2D domain,
thermalization only reflects variance in the tangential component of the
shuffle vectors, not the radial component. Different types of noise can be fit
to the MD data.

###  3.6 Displacement pole figures: implementation and conventions

The thermalized displacement texture is estimated directly from MD data via
probability density estimation. The information contained in the displacement
texture can be separated into shuffle length and orientation distributions. In
this work, we pursue a visualization method for shuffle orientation
distributions that optionally includes information about shuffle length.
Displacement pole figures plot displacement orientations in stereographic
projection relative to the sample frame of the bicrystal. The grain boundary
orientation in the inverse pole figures is always the same; displacement
vectors lying in the GB plane (for a flat GB) plot along vertical lines in the
stereographic projections of the upper and lower hemisphere. Displacement pole
figures are generated in MTEX with a spherical harmonic fit with the default
de la Vallee-Poussin kernel [53]. Displacement magnitude may be added to the
spherical harmonic fit as an additional weighting factor for each atom. In
this work, displacement pole figures plotted in the TDP and CDP contain
displacement length information, while displacement pole figures plotted in
the OTDP contain displacement orientation information only. The weighted
displacement pole figures are convenient for visualizing large, low
probability displacements which are commonly associated with multi-hop GB
diffusion events.

Displacement pole figures provide a convenient starting point for counting and
classifying shuffle types during low $\Sigma$ (typically $\Sigma<13$) GB
migration and give insight into nearly continuous shuffle orientation
distributions during high $\Sigma$ GB migration. In cases where distinct
displacement poles are well separated in orientation space ($>10^{\circ}$), a
probability threshold is defined to count the number of unique displacement
types. In this work, we determine a unique shuffle type as one corresponding
to a maximum pole probability in the OTDP greater than 0.005 with mean
displacement length larger than typical atomic vibrations ($>0.04$ nm). For
high $\Sigma$ GBs or high temperature GB migration, displacement length
information must be incorporated into counting routines since distinct
shuffles may exist with closely spaced orientations or significantly
overlapping orientation distributions. For most accurate shuffle
identification, shuffles enumerated from the forward model can be directly
compared to the MD data at varying degrees of regularization as in [37]. We
recommend that comparisons in noisy settings be performed in the OTDP.

##  4 Applications of displacement texture

###  4.1 Characterization of $\Sigma 3$ twin boundary migration in FCC Ni

In this section, we show that the boundary plane inclination variation of
shuffling patterns for $\Sigma 3$ GBs in MD simulations takes a simple form
that can be rationalized by distance minimization in the dichromatic pattern
with respect to microscopic translations. This connection between microscopic
translational degrees of freedom and boundary plane dependent shuffling
patterns is found to be generic to all low $\Sigma$ GBs in our MD datasets for
FCC Ni.

Twin GBs with a $\Sigma 3$ misorientation are common in FCC materials and
exhibit anisotropic mobility behavior depending on boundary plane inclination
[66, 67]. The most common types of $\Sigma 3$ GBs in FCC metals are the
$\hkl(111)$ Coherent Twin Boundary (CTB) and the $\hkl(112)$ Incoherent Twin
Boundary (ITB). The acronym ITB refers to any facet that is not $\hkl(111)$
type. ITBs form as orthogonal or near orthogonal facets along CTBs. Flat CTBs
are known to be extremely immobile, and are the basis for many GB engineering
efforts because of their low grain boundary energy, high yield strength and
high thermal stability [6, 68, 3]. On the other hand, ITBs are highly mobile
and ITB facet motion is observed during diverse processes such as
recrystallization, grain growth, radiation damage and stress driven coarsening
[10, 69].

####  4.1.1 The importance of microscopic shifts in determining shuffling for
different plane inclinations

Only three distinct types of shuffling transformations are observed for 41
$\Sigma 3$ GBs in our dataset over a relatively large temperature range from
100-1000 K under both free and constrained boundary conditions. Most $\Sigma
3$ GBs move with zero net shear, with two exceptions (the (112) ITB and (111)
CTB) discussed in Section 4.1.2. Displacement pole figures corresponding to
migration are plotted in Figure 7) for different boundary inclinations at 300
K. We define characteristic displacement textures as shuffling patterns in the
OTDP that are unique up to a rigid rotation. Characteristic displacement
textures are observed with three, four and six unique displacement poles.

The six pole transformation is only observed for the CTB. In contrast, the
four pole transformation is observed for a sequence of $\hkl<110>$ asymmetric
tilt GBs that move via glide of $\hkl(112)$ type ITB facets. The most common
transformation in the MD data has only three poles and is observed during
$\hkl(110)$ ITB facet glide. The rigid rotation of poles with GB geometry
simply represents the spatial rotation of facets that control migration of
these GBs.

One target application of displacement texture analysis is the
crystallographic identification of shuffling vectors in the crystal frame of
each grain. Recall that the choice of microscopic shift $\bm{p}$ impacts
shuffling directions. When shuffling directions are expressed in the crystal
frame of the CDP, as shown in Figure 8, they are most interpretable with
respect to prior literature. The six pole mechanism has six Shockley partial
vectors along $\hkl<112>$ Shockley partial directions known to contribute to
CTB motion. The four pole mechanism has two $\hkl<112>$ type and two
$\hkl<110>$ type directions, consistent with slip vector analysis in [24]. The
three pole mechanisms has two vectors pointing along $\hkl<112>$ directions
and one along the $\hkl<110>$ direction (with magnitude smaller than a full
dislocation), consistent with a picture of boundary migration mediated by
triplets of partial dislocations [66]. Prior slip vector analysis for the
$(110)$ ITB in [24] originally reported a mechanism with nine displacement
types, but this analysis was later amended to a three shuffle mechanism in
[25] upon more careful analysis of thermal noise.

Exhaustive enumeration of shuffling patterns for $\Sigma 3$ GBs via the
forward model in [37] (without shear coupling and at small regularization)
showed four distinct classes of transformations with 3, 4, 6 and 7 unique
displacement poles for different microscopic translations. The first three
types of shuffling patterns are found in our MD data and the relative
frequency of these transformations in inclination space matches the frequency
in translation space. For instance, a randomly chosen plane inclination or
microscopic shift is most likely to yield a 3 shuffle distance minimizing
transformation. If the microscopic shift $\bm{p}$ is chosen explicitly in the
dichromatic pattern on the basis of energy minimization (as in Fig 8), the
displacement patterns enumerated by the forward model match the MD data
exactly.

The correspondence between the distance minimizing shuffling pattern selected
for different microscopic shifts and the shuffling pattern selected for
different plane inclinations has important implications for modeling grain
boundary mobility. The problem of determining the dependence of mobility on
plane inclination can be approximated as the problem of determining the
energetics of a discrete set of shuffling patterns which vary with microscopic
shift. Different values of $\bm{p}$ which are predicted to correspond to
distinct transformations can be used to seed bicrystal simulations or NEB
calculations. For each characteristic displacement texture, there is a
distinguished choice of $\bm{p}$ which minimizes total shuffling distance (and
defines the OTDP). It is hypothesized that this choice of shift vector locally
maximizes mobility within a characteristic displacement texture class.

Characteristic displacement textures are observed for all other
misorientations with sufficient inclination sampling in the FCC Ni dataset
(Figure 11). $\Sigma 5-11$ GBs provide a rich and complex set of migration
mechanisms for further analysis, but are only pursued in this work to the
extent that they illustrate new concepts. Shear coupling behavior and thermal
behavior of selected $\Sigma 5$ GBs is discussed in Section 4.2.2. The
multimodal migration of a $\Sigma 11$ GB is discussed in Section 4.3.1, where
it is shown that choice of $\bm{p}$ also impacts the relative amount of GB
diffusion and shuffling expected during migration.

Figure 7: For $\Sigma 3$ GBs with varying GB plane inclination, multiple types
of shuffling transformations are observed. Points in the boundary plane
fundamental zone are colored by the number of unique shuffle vectors observed
in the displacement texture of each GB at 300 K. Displacement pole figures in
the OTDP illustrate three classes of transformations corresponding to 110
facet (3 shuffle), 112 facet (3-4 shuffle) or 111 facet (6 shuffle) controlled
motion. Figure 8: Min shuffle transformations for $\Sigma 3$ twin GBs in the
TDP and corresponding displacement pole figures in the CDP. Degenerate
displacements correspond to equal length displacements.

####  4.1.2 Degenerate displacements and implications for period doubling and
sliding

Displacement texture aids the interpretation of the thermal and mechanical
behavior of $\Sigma 3$ GBs. An important aspect of the four and six pole
transformations for the $\hkl(112)$ ITB and $\hkl(111)$ CTB is that they
contain degenerate length displacements in the dichromatic pattern. According
to the geometry based forward model, an atom is equally likely to choose any
of the degenerate displacements during migration. However, in reality, during
migration, once the degeneracy in the shuffling pattern has been broken (by a
fluctuation or small external perturbation), the remainder of the atomic
displacement must proceed in a way that maintains compatibility of the net
reorientation. In the MD data, an interesting period doubling effect is
observed for $\hkl(112)$ ITB motion in which every tenth plane of shuffles
alternates between degenerate displacements of opposite parity in the 4 pole
transformation (purple and green vectors in Figure 9). It is expected that one
displacement type could be favored with strains which break the degeneracy in
the dichromatic pattern.

Figure 9: Displacement texture for $\Sigma 3$ $\hkl(811)||\hkl(554)$
asymmetric tilt GB at 300 K demonstrates the 4 pole transformation mediated by
112 ITB facet migration. The length degeneracy in the displacement pattern
results in a period doubling effect where purple and green displacements
alternate orientation along layers of (111) planes. An orientation change
within a single plane is highlighted. Such period doubling effects are common
across the FCC Ni dataset and are sensitive to microscopic shift $\bm{p}$.

The degeneracy of the six pole transformation for the CTB has implications for
stress generation during migration. Three degenerate twinning partials are
available on two of three parallel $\hkl(111)$ planes leading to six unique
displacement directions total for sliding (Figure 8). As described by Thomas
in [70], all disconnection modes ($\bm{b}$,$\bm{h}$) for the CTB can be
obtained from integer multiples of two burgers vectors
$\bm{b_{1}}=\hkl[1-12]a/6$ and $\bm{b_{2}}=\hkl[-211]a/6$ (where $a$ is the
lattice parameter) and integer multiples of step heights
$\bm{h_{10}}=-\hkl[-111]a/3$. The simplest type of disconnection motion
involves only one displacement given by one of three twinning partials
($\bm{b_{1}}$, $\bm{b_{2}}$ or
$\bm{b_{3}}=-\frac{1}{2}(\bm{b_{1}}+\bm{b_{2}}))$ and has step height
$\bm{h_{10}}$ associated with the spacing between neighboring $\hkl(111)$
planes. This type of motion is favored by a strain along $\bm{b_{1}}$,
$\bm{b_{2}}$ or $\bm{b_{3}}$. It is an example of a pure shear transformation
with no shuffles. Energy jump driven migration of the CTB in our simulations
is associated with only a small amount of net shear, even for significant
normal motion. The low coupling factor of this transformation, combined with
the presence of all six displacements, indicates an alternating mixture of
sliding displacements during migration. Thus, even though the GB may move with
little apparent shear, it generates a shear stress and strain in its
transition state. This behavior is contrary to the prediction of the
disconnection model that energy jump driven motion of the CTB corresponds to
pure step motion. Pure step motion is not possible because of the sliding
degeneracy in the min-shuffle mapping. Stresses are generated during
constrained migration of the CTB despite the zero net shear character of
motion observed in many of our simulations. Such a situation involving
alternating sliding events is likely common during zero shear migration of
ordered GBs [28] and has been reported in atomistic simulations which show
anomalous stress driven motion of $\Sigma 7$ GBs with zero net strain [71,
72]. We note that alternating sliding events which result in migration with
zero apparent shear may be coupled to shuffling (with finite coupling factor)
or may have pure sliding character (infinite coupling factor).

The $\Sigma 3\hkl(112)$ ITB generates a stress under constraints for different
reasons: zero shear migration is in competition with partial dislocation
emission, a process associated with a small shear strain [28]. At sufficiently
low temperatures with free boundary conditions ($T<900$ K) and a ramped energy
jump driving force, the $\hkl(112)$ ITB emits stacking faults into the grain
ahead of the interface, resulting in a net shear. In the constrained case,
this shear generates a back stress which arrests further dislocation
extension. Migration is then promoted as driving force is ramped upward.
Displacement poles may deviate from their ideal positions in the
characteristic displacement texture with changing length of emitted
dislocation and with changes in the partial dislocation structure of the GB.
Extended defects may point outward or inward from the GB plane depending on
temperature. The OTDP is a convenient representation in this case because
small strains that arise from changes in boundary structure and dislocation
emission are subtracted out of the displacement texture. The displacement pole
locations in the OTDP vary little with temperature from 100 to 1000 K, though
the relative probabilities of degenerate displacements may vary.

####  4.1.3 Quantifying thermal noise for high mobility migration of the
$\Sigma 3\hkl(110)$ GB

Figure 10: Displacement texture analysis allows for quantification of thermal
behavior. (a/b) Displacement poles for $\Sigma 3$ (110) GB broaden with
temperature, as shown by a decrease in maximum pole probabilities. (c)
Displacement lengths also tend to increase with temperature. The analysis is
performed in the OTDP to regularize over small shifts in displacement fields
arising from the Shockley partial structure of the GB.

To illustrate the detailed information that can be extracted from displacement
textures, we characterize the thermal behavior of the highly mobile $\Sigma 3$
(110) ITB at temperatures from 100-1400 K. The three pole transformation
remains stable at high temperatures and the probability mass around each pole
broadens (Figure 10a). One way to quantify thermal broadening of the
displacement texture is by tracking the maximum probability of each pole in
the spherical harmonic fit. Displacement length distributions can also be
computed for the displacements corresponding to each pole. It is observed that
the angular deviation of shuffles and the average length of shuffles generally
increase with temperature, matching intuition. However, the average increase
in shuffle length and the decrease in peak probability are not necessarily
monotonic. Small changes in the Shockley partial structure of the boundary may
impact thermal broadening of the displacement texture. The probability decay
is not well modeled by a gaussian solution to the diffusion equation.

###  4.2 Characterizing the misorientation dependence of displacement texture

In this section, we characterize displacement textures for mobile GBs at a
variety of misorientations including both low and high $\Sigma$ GBs. At 300 K,
over a third (142/388) of the GBs in the FCC Ni dataset move with near zero
shear ($|\beta|<0.05$) under unconstrained boundary conditions and a ramped
energy jump driving force [45]. We begin by characterizing the motion of low
$\Sigma$ GBs which move with zero net shear. The correspondence between plane
inclination and microscopic shift for $\Sigma 3$ GBs is found to generalize to
$\Sigma 5-11$ GBs. Next, we show that shuffling patterns may retain a high
degree of order even for high $\Sigma$ GBs. We discuss common features of
displacement textures for high $\Sigma$ GBs such as vortex displacement
patterns.

####  4.2.1 Characteristic shuffling patterns for low $\Sigma$ GBs

Different characteristic displacement textures are observed for distinct
misorientations in the FCC Ni dataset depending on boundary plane inclination.
The number of unique displacement poles (tabulated for constrained migration
of low $\Sigma$ GBs at 300 K and shown in Figure 11) is found to be a useful
identifier of distinct characteristic displacement textures. For each
misorientation, a minimal characteristic displacement texture is observed with
the fewest shuffle displacement types. The minimum number of shuffles is found
to equal $\Sigma$ for $\Sigma 3$ GBs and $\Sigma-1$ for $\Sigma 5-11$ GBs.
These values can be rationalized by the CSL geometry of the CDP. If one atom
in every CSL unit cell is coincident, $\Sigma-1$ displacement types are
required for reorientation. However, depending on crystal structure, the CDP
may have degenerate length shuffles which make observation of more than
$\Sigma-1$ shuffles likely. This is the case for $\Sigma 3$ GBs, where the
zero shear transformation in the CDP is associated with the six shuffle
mechanisms discussed for the (111) CTB. In $\Sigma 3$ GBs in BCC Fe, such
degeneracy is not present in the CDP and 2-shuffle transformations are common.
Depending on microscopic shift, different types of degeneracies in shuffling
patterns may arise such that (min(# poles ) $>$ $\Sigma-1$). These boundary
plane dependent shuffling transformations are observed for all low $\Sigma$
GBs in Figure 11 except $\Sigma 7$ GBs. We note that some GBs that have a
large number of shuffles may also contain non-optimal displacements (as
discussed in Sections 4.2.2, 4.2.3 and 4.3).

Shear coupling changes the expected shuffling geometry in the dichromatic
pattern and can reduce the number of shuffles required for reorientation to
well below $\Sigma-1$ and even zero in the case of a pure shear
transformation. For example, the shuffling pattern enumerated in the forward
model for a $\Sigma 15\hkl<112>$ tilt GB has only four unique shuffle vectors
(Figure 15), fewer than the 14 expected on the basis of analysis of zero shear
transformations.

Figure 11: Distinct characteristic displacement textures at fixed
misorientation can be separated into classes by the number of unique shuffle
displacement types in the displacement pole figure. Point size is proportional
to the number of boundaries within each class. Figure 12: Distinct zero shear
characteristic displacement textures at fixed misorientation can be separated
into classes by the number of unique shuffle displacement types in the
displacement pole figure. Plotted squares denote boundaries which sweep out
shear during motion and are excluded from the current analysis.

In light of the analysis of $\Sigma 3$ GBs in Section 4.1, we hypothesize that
the distinct classes of shuffling transformations that arise for different
plane inclinations in $\Sigma 5-11$ GBs can be explained by the impact of
microscopic shifts on distance minimizing mappings in the dichromatic pattern.
The forward model may be used to test this hypothesis by exhaustively
enumerating characteristic displacement textures with respect to shift vector
and regularization and comparing to MD data. In the next section, we compare
$\Sigma 5$ displacement textures observed in MD data to predictions of the
forward model. In Section 4.3.1, this analysis is extended to better
understanding the rigid sliding and migration of a $\Sigma 11$ twist GB.

####  4.2.2 Characteristic shuffling patterns for $\Sigma$5 GBs

$\Sigma 5$ GBs exhibit diverse shuffling behavior at 300 K with at least four
distinct transformation types observed under constrained boundary conditions
with varying plane inclination (Figure 12). The four pole transformation
corresponds to the $\Sigma 5\hkl(100)$ twist GB and $\Sigma 5$ GBs with nearby
inclinations. This four-shuffle mechanism was observed in early MD studies of
curvature driven GB migration by Jahn and Bristowe [73] and is optimal in the
sense that it minimizes net shuffling distance in the TDP [37]. Five, six and
seven pole transformations are also observed for distinct inclinations (Figure
12). Several GBs (marked with squares in Figure 12) slide significantly even
under constrained boundary conditions.

Figure 13: Three types of shuffling patterns observed for $\Sigma 5$ GBs with
zero net shear deformation and containing (a) 4, (b) 5 and (c) 7 or more
nonzero poles. All snapshots are shown at 500 K. Figure 14: Enumeration of
shuffling patterns for $\Sigma 5$ GBs in forward model with
$\bm{F^{GB}}=\bm{I}$, $\epsilon=0.005$ and varying microscopic shift $\bm{p}$.
(a) Grid search over shift vector in DSC lattice reveals several distinct
transformations, conveniently parametrized by number of unique nonzero
displacements. (b) Transformations depend on the choice of $p$ within the
search space in distinct ways. (c) Three example transformations are shown.
The last transformation is particularly significant because of its appearance
in both the roughened migration of the $\Sigma 5$ (310) GB [37] and zero shear
migration of asymmetric tilt GBs.

Non-minimal zero shear characteristic displacement textures for $\Sigma 5$ GBs
may be rationalized by the impact of the microscopic shift vector on allowed
mappings in the TDP, similar to prior analysis for $\Sigma 3$ GBs. The forward
model is used to enumerate shuffling patterns for 1000 shift vectors in the
$\Sigma 5$ DSC unit cell with $\bm{F^{GB}}=\bm{I}$. At least four distinct
transformations are observed across both the MD data and the forward model.
Transformations from the forward model and MD data show striking similarities.
For instance, the 8 pole transformation in Figure 14, which arises from
degenerate displacements in the TDP, appears to match the displacement texture
in Figure 13 for a $\Sigma 5$ asymmetric tilt GB up to a rigid rotation.
Interestingly, a very similar pattern was observed during roughened migration
of the $\Sigma 5$ (310) GB at high temperatures in [37], where it is argued
that the pattern arises from a combination of disconnection modes of opposite
sign along with longer, non-optimal vectors. The forward model offers a way to
simplify and explore the rich phase space of GB motion. The two highest
displacement numbers in the forward model $(7,8)$ and the MD data $(6,7)$ do
not match. This mismatch arises from how poles in the MD data are counted in
the presence of noise. The unique pole counts tabulated from the MD data
should be viewed as a lower bound for the true number of unique displacement
types because 1) poles must be distinguished within an angular cutoff in the
MD data and 2) low probability displacement types below a threshold
probability are ignored. With further work, such as the templating method
described in [37], displacement textures from MD data and the forward model
can be compared more quantitatively.

####  4.2.3 Characteristic shuffling patterns for high $\Sigma$ GBs

In this work, a GB is called high $\Sigma$ if local maxima in the displacement
texture are sufficiently close in angle to form continuous bands in the
displacement pole figure. This typically occurs for $\Sigma>19$ within our
dataset. Previously, it has been assumed that high $\Sigma$ GBs cannot move
via the ordered shuffling transformations of low $\Sigma$ GBs because of the
complex shuffles required in large CSL unit cells ([74]). The FCC Ni migration
dataset offers several counterexamples to this statement, but also highlights
the importance of simultaneous GB diffusion and migration mechanisms during
the motion of high $\Sigma$ GBs.

Two ubiquitous ordered displacement morphologies during high $\Sigma$ GB
migration include displacement vortices and displacement helices. These
patterns often arise during shear coupled migration of symmetric tilt GBs
(Figure 15). Vortices and helices accommodate a net shear deformation and can
be decomposed into ordered layers of shuffles and pure shear displacements.
Vortices may also be present in systems with zero net shear deformation, such
as twist GBs. In twist GBs, displacement vortices are confined to the
interface plane, circulating around an axis $\hat{\bm{v}}$ pointing in the GB
normal direction. In symmetric tilt GBs, vortices circulate around the tilt
axis $\hat{\bm{v}}$ in planes perpendicular to the GB plane. In asymmetric
tilt and mixed tilt/twist GBs, vortices may lie along planes inclined to the
GB. Interestingly, the length scale of vortex displacement patterns is
sensitive to geometric confinement. Basketweave displacement textures are
observed during the constrained migration of symmetric tilt GBs in which
vortices tile space with zero net shear. These patterns, discussed further in
Section 4.3.2, are hypothesized to arise from a primary disconnection mode
mixed with one or more mechanisms that reduce the net shear deformation, such
as as GB diffusion coupled sliding or secondary disconnection motion.

Figure 15: The displacement texture of the $\Sigma 15$ ¡211¿ $48.19^{\circ}$
tilt GB can be decomposed into a set of layered shuffles (middle) and a shear
deformation associated with a zero shear reference angle of $0^{\circ}$. The
helical displacement texture (left and right panels for predictions of MD and
the forward model) results from the superposition of shuffles and shears.

It is demonstrated via the forward model that the ordered layers of shuffles
extracted from the shear coupled migration trajectories in Figure 15 are
optimal in the sense that the shuffles minimize net transformation distance in
the TDP. In the FCC Ni dataset, unimodal shear coupled migration at low and
intermediate temperatures (100-1000 K) satisfies the min-shuffle hypothesis.
Nevertheless, a small fraction of non-optimal displacements may be present
that grow in probability with increasing driving force or temperature. Non-
optimal displacements correspond to competing atomic rearrangement mechanisms.
For instance, the $\Sigma 89$ ¡100¿ $26^{\circ}$ tilt GB exhibits diffusional
flow along the tilt axis during migration that increases in probability with
temperature. The displacements along the tilt axis appear in the displacement
pole figure as poles isolated from the continuous band of vortex displacements
(Figure 16). Depending on GB geometry (both microscopic and macroscopic
degrees of freedom $\bm{p}$), non-optimal displacements may form networks.
Large, non-optimal displacements of the $\Sigma 85$ (100) twist GB were
previously found to form rectangular arrays within the plane of the GB [37]
and are thought to be associated with GB self diffusion in screw dislocation
cores. Non-optimal displacements were localized to regions where minimum
length shuffles were large. The $\Sigma 85$ (100) twist GB is an example of a
GB in the FCC Ni dataset that always has a relatively high fraction of non-
optimal displacements, even at low and intermediate temperatures. Because of
the long shuffles necessary for geometric compatibility, and the likelihood of
many pathways of similar length, many high $\Sigma$ GBs are dissimilar from
special low $\Sigma$ GBs which can move via athermal shuffling mechanisms that
do not involve diffusion. Shear coupling reduces the complexity of shuffles
but may still involve significant amounts of GB diffusion, depending on
microscopic shift, temperature and driving force.

Figure 16: Displacement pole figures for high $\Sigma$ GBs of varying GB type
(ST = symmetric tilt, AT = asymmetric tilt, TW = twist, Mixed = mixed
tilt/twist). Arrows on displacement pole figures show the planar cut that is
used to visualize the shuffling patterns. Cross sections are perpendicular to
the tilt axis and in the plane of the boundary for ST and TW GBs. GBs are ST:
$\Sigma 89$ ¡100¿ 28∘ (ID 264), TW: $\Sigma 235\hkl(310)$ (ID 369), AT:
$\Sigma 51\hkl(521)$ (ID 362), Mixed: $\Sigma 81\hkl(841)||\hkl(744)$ (ID
273), where IDs are the same as in [13].

###  4.3 Characterizing nonuniform and roughened migration

During grain growth, boundaries do not migrate uniformly. Migration is often
observed to be a jerky, stochastic process associated with changes in local
driving forces and boundary conditions. It is important to gain a better
fundamental understanding of nonuniform GB migration. A counterpart to
nonuniform migration is roughened migration. Above a critical roughening
transition temperature that depends on GB type and active migration mechanism,
GB mobility jumps from near zero to a larger value. A recent disconnection
based model for roughening points to the dual roles of temperature and driving
force in facilitating roughening ([64]). High driving forces (energy jumps or
stresses) lower the effective roughening temperature. During ramped driving
force MD simulations of bicrystals, transitions in migration behavior are
observed which depend on GB type, temperature, driving force and boundary
conditions [45]. In this section, we employ ramped driving force simulations
in bicrystals with constrained boundary conditions. With this simulation
setup, GBs which yield with sliding character undergo up to two regimes of
motion: 1) an initial slow phase of motion characterized by stair-steps in the
GB position and average shear stress and 2) a fast period of continuous
runaway motion that accommodates zero shear boundary conditions. We denote the
driving force at the initial onset of motion as the critical driving force and
the driving force at the onset of runaway motion the second critical driving
force. A mechanistic account of roughening at the second critical driving
force is a main contribution of this section.

Two types of nonuniform migration will be examined in this section. First, we
will explore the sequential sliding and migration of a $\Sigma 11$ twist GB.
This GB is representative of a poorly understood behavior for twist GBs
observed in a recent MD survey [45] in which GBs rigidly slide by a small
increment before moving forward. $\Sigma 11$ GB motion has also recently been
analyzed in the context of asymmetric motion in which a specific GB was found
to have different mobilities in the forward and backward direction under
energy jump driving forces of opposite sign [75, 76]. In Section 4.3.2, we
explore pattern formation during roughened migration of symmetric tilt GBs
upon second yield. An open question in the literature concerns the mechanisms
by which GBs lose their shear coupling factor at high temperatures or under
constraints. Zig-zag motion has been observed in a specific case [36], but it
is not well understood whether zig-zag motion is generic to roughened
migration or whether other mixed mechanisms such as diffusion assisted sliding
and coupling are possible. We show that characteristic tiled vortex patterns
during constrained migration of tilt boundaries arise from a combination of
disconnection motion, sliding and GB diffusion. Depending on GB geometry, zig-
zag motion may also be observed. These examples demonstrate how displacement
texture analysis, combined with a forward model for migration, assist the
interpretation of multimodal migration data.

####  4.3.1 Nonuniform sliding and migration of a $\Sigma 11$ twist GB

The $\Sigma 11$ (110) GB offers an example of multimodal migration that
involves rigid sliding at the beginning of motion followed by normal
migration. In combination with the forward model, displacement texture
analysis gives insight into shuffling geometry during this type of motion. At
least 17 unique shuffles are present, indicating a non-minimal displacement
texture. Bands of displacements are visually apparent with relatively large
vertical components along the Y direction (Figure 17). Ordered layers of
shuffles lie in between these bands with components primarily in the XZ plane.
The displacement pole figure in Figure 17d distinguishes between the
displacements in the bands and the displacements between bands along the
equator.

This displacement geometry can be rationalized on the basis of the forward
model only if the impact of sliding on microscopic degrees of freedom is taken
into account. Additionally, regularization in the forward model is necessary
to approximate non-optimal displacements observed in the MD data. Unlike
previous examples, the forward model does not reproduce the observed
displacement pattern at any $\epsilon$ when $\bm{p}$ is chosen to match the
relaxed 0 K bicrystal structure. We hypothesize that a different choice of
$\bm{p}$ leads to a matching displacement pattern for some $\epsilon$. A
random search over 100 $\bm{p}$ grid points at $\epsilon=0.25$ reveals a
candidate displacement pattern from the forward model as shown in Figure
17b-d. This purely geometric prediction captures the banded pattern shown in
the MD data but misses some displacements within the bands compared to the MD
data. As the forward model does not capture the details of metastable GB
structure and only approximates the net result of multi-hop sequences of
displacements, it is unsurprising that it does not predict all non-optimal
displacements in the MD data. Nevertheless, it has been shown that alternating
shuffling and sliding events accommodate constrained migration of the $\Sigma
11$ (110) GB. The forward model gives useful insight into the impact of $p$ on
allowed mappings in the dichromatic pattern.

Figure 17: (a) Displacement texture in TDP for $\Sigma 11$ (110) twist GB
demonstrates banded displacement pattern. Because a sliding event occurs at
the onset of motion, the TDP is modified from the initial state and is treated
as unknown. (b) Shuffling patterns are enumerated for 100 random shifts
$\bm{p}$ at relatively large regularization $\epsilon=0.25$. (c/d) The best
match to MD data is found to be a 16 pole transformation. Though the layered
structure is apparent in the prediction of the forward model, the MD data
clearly contains more types of displacements. This example shows the general
strategy of optimizing over $\bm{p}$ and $\epsilon$ in order to match the MD
data.

####  4.3.2 Constrained migration mechanisms of symmetric tilt GBs

Constrained migration of symmetric tilt GBs often involves basketweave
displacement textures with tiled displacement vortices and helices. See, as
examples, Figure 1 (top left) and Figure 16 (top). We hypothesize that these
patterns can be understood as arising from a primary disconnection mode mixed
with a mechanism that reduce the net shear deformation, such as as GB
diffusion mediated sliding or secondary disconnection nucleation and motion.
Displacement texture analysis aids the interpretation of these complex
multimodal migration scenarios.

In this section, we analyze roughened migration of the $\Sigma 5$ (310) GB. At
temperatures above the roughening temperature, this GB is observed to move
with zero net shear regardless of boundary conditions [77, 37]. Since $\Sigma$
is small, distinct displacements can still be resolved in the displacement
pole figure (Figure 18b). The displacement texture is not minimal and has at
least 9 unique nonzero displacements. We attempt to infer the mixture of
migration mechanisms from the mixture of displacements in the pole figure. Two
candidate mixtures are 1) a laminated combination of two disconnection modes
or 2) a zero shear transformation with regularization corresponding to GB
diffusion like displacements.

Figure 18: Comparison of displacement textures for (b) roughened migration of
$\Sigma 5$ (310) GB at 800 K under constrained boundary conditions and (a)
zero shear transformation enumerated from the forward model with 9 unique
poles (regularized version of 8 pole transformation in Figure 13).

Displacements corresponding to shear coupled motion of the $\Sigma 5$ (310) GB
in the positive and negative $\hkl(100)$ and $\hkl(110)$ type coupling modes
were previously enumerated at small $\epsilon$ [37]. It was found that the
roughened displacement pattern contains displacements from both types of
disconnection modes and that shuffles for different modes mix in the plane of
the GB, rather than in a zig-zag fashion. However, the mixture predicted from
the forward model does not exactly match the MD simulations. Displacements
along the tilt axis are observed in the MD data but are not predicted for
individual disconnection modes in the forward model [37], even at
$\epsilon=0.25$. On the other hand, a zero shear displacement texture
predicted from the forward model with appropriately chosen $\bm{p}$ and
$\epsilon=0.25$ captures at least six of the nine poles in the MD data,
including displacements along the tilt axis. The agreement is not perfect and
the zero shear transformation appears to miss several vortex related
displacements in the MD data which are captured in the disconnection shuffling
predictions. In summary, both of the candidate mixtures of mechanisms from the
forward model show partial agreement with the displacement texture from the MD
data. We cannot rule out the possibility that there is a different combination
of $\bm{p}$ and $\epsilon$ (possible related to microsliding events during
migration) in the forward model that give better agreement to the MD data.
Alternatively, the simplified distance based potential in the forward model
may be insufficient to capture the complexity of the MD simulation, which
capture a combination of GB diffusion and disconnection based shuffling with
statistics different from the forward model.

##  5 Summary and Conclusions

Shuffling patterns during GB migration strongly depend on bicrystallography,
even in complex multimodal migration scenarios. This work provides a
simplified modeling and characterization framework for rationalizing the
impact of GB geometry in the dichromatic pattern on observed atomic
rearrangements during GB motion. High throughput MD simulations in FCC Ni at
low and intermediate temperatures reveal a rich variety of GB migration
mechanisms which include competing shuffling modes and mixtures of shear
coupling, sliding and GB self diffusion.

The regularized optimal transport based forward model for migration presented
in [37] is shown to significantly clarify analysis of the many mechanisms
observed in MD simulations. The forward model approximates GB migration as a
probability distribution over shuffling transformations in the dichromatic
pattern. In the forward model, the optimal transformation minimizes net
shuffling distance and competes with more costly transformations at finite
regularization. Net shuffling cost may be minimized over multiple candidate
disconnection modes associated with shear deformations in the dichromatic
pattern. Main scientific contributions of our MD survey, in conjunction with
analysis from the forward model, are as follows:

  1. 1.

At low and intermediate temperatures, the reorientation geometry of many low
$\Sigma$ GBs, including those that shear couple, is dominated by a small set
of optimal shuffles which are distance minimizing in the dichromatic pattern.
Shear coupling reduces the number of shuffles required for conservative GB
migration relative to zero shear transformations.

  2. 2.

Changes in microscopic degrees of freedom impact shuffling patterns in
significant and quantifiable ways. The boundary plane dependence of shuffling
geometry for low $\Sigma$ GBs can be explained by the impact of microscopic
translations on distance minimizing mappings in the dichromatic pattern. Small
sliding events during migration or changes in GB character may change the
dominant shuffling mechanism.

  3. 3.

Period doubling effects are discovered in the displacement patterns of GBs
with degenerate length shuffles available in the dichromatic pattern, such as
the $\Sigma 3$ (112) GB and the $\Sigma 3$ (111) GB. In the latter case,
degenerate displacements manifest as well known sliding events associated with
partial dislocations.

  4. 4.

Non-optimal shuffles, often associated with stringlike displacements in the MD
data, are found to concentrate in regions where many shuffles of nearly
equivalent length can occur (such as the $\Sigma 11$ GB in Figure 16a). When
they lie in the plane of the GB, these displacements are associated with GB
diffusion. Non-optimal displacements are common in high $\Sigma$ GBs.

  5. 5.

Vortex displacement patterns are ubiquitous during shear coupled motion and
tilings of vortices appear to be signatures of constrained migration of ¡100¿
and ¡111¿ symmetric tilt grain boundaries in FCC Ni. The exact combination of
mechanisms contributing to these vortex tilings is unclear, with optimal zig-
zag motion insufficient to describe all observed displacements.

There are a variety of opportunities to improve upon and extend this work to
gain a better scientific understanding of GB migration. One limitation of the
displacement texture analysis in this work is the coarse time resolution
employed (typically $\Delta t=0.1$ ns before and after migration). Analysis of
characteristic timescales associated with shuffling and dynamical
heterogeneity would be beneficial at both shorter and larger migration times
to identify rate limiting mechanisms associated with migration and to extend
analysis of unit mechanisms to higher temperatures.

Displacement texture analysis can be extended to other MD datasets to analyze
unit mechanisms in complex settings such as curved GB migration,
polycrystalline grain growth and GB migration in alloys. Analysis of
displacement textures for different metastable GB phases would be particularly
interesting to directly probe the impact of microscopic degrees of freedom on
shuffling mechanisms. Using the methods in this work, shuffling mechanisms can
be systematically enumerated for any CSL GB and crystal structure for varying
microscopic shifts and levels of regularization. Both the forward model and
displacement texture analysis could help infer migration mechanisms from HRTEM
images taken during migration.

## Acknowledgments

Work at CMU was supported by National Science Foundation grants GFRP-1252522
and DMR-1710186. The authors gratefully acknowledge feedback from Ankit Gupta,
John Hirth, Yuri Mishin, Greg Rohrer, Tony Rollett and Garritt Tucker that
helped improve the work.

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URL
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## Supplementary Information

##  Appendix A Determination of C-B deformation

In this section we present a compact algorithm for computing the Cauchy-Borne
deformation associated with the shear-coupled deformation of a N-atom lattice
due to GB motion. $\\{\bm{X}_{1},\ldots,\bm{X}_{N}\\}$ and
$\\{\bm{x}_{1},\ldots,\bm{x}_{N}\\}$ are the initial and final positions of
atoms in the crystal, respectively, both of which are known in the context of
molecular dynamics analysis. We approximate the corresponding affine
deformation gradient as the solution to the following optimization problem

$\displaystyle\bm{F}^{GB},\bm{u}_{0}=\underset{\bm{F},\bm{u}_{0}}{\operatorname{\arg}\inf}\frac{1}{p}\sum_{i}\Big{|}(\bm{F}\bm{X}_{i}+\bm{u}_{0})-\bm{x}_{i}\Big{|}^{p}$
(10)

where $p\geq 1$. For the special case of $p=2$, the stationarity condition is

$\displaystyle\bm{F}^{GB}$
$\displaystyle=\Big{(}\sum_{m}(\bm{x}_{m}-\bm{u}^{0})\otimes\bm{X}_{m}\Big{)}\Big{(}\sum_{n}\bm{X}_{n}\otimes\bm{X}_{n}\Big{)}^{-1},$
$\displaystyle\bm{u}_{0}$
$\displaystyle=\frac{1}{N}\sum_{n}(\bm{x}_{n}-\bm{F}\bm{X}_{n}).$ (11)

###### Proof.

We now switch to index notation. Subscript indices follow the summation
convention and are summed over spatial dimensions. Superscript indices are
only summed explicitly, and correspond to atomic enumeration:

$\displaystyle\bm{F}^{GB},\bm{u}_{0}=\underset{\bm{F},\bm{u}_{0}}{\operatorname{\arg}\inf}\frac{1}{2}\sum_{n}[(F_{pQ}X_{Q}^{n}+u^{0}_{p})-x_{p}^{n}][(F_{pR}X_{R}^{n}+u^{0}_{p})-x_{p}^{n}]$
(12)

The Euler-Lagrange for $\bm{F}^{GB}$ follows by straightforward calculation:

$\displaystyle 0$ $\displaystyle\overset{!}{=}\frac{\partial}{\partial
F_{iJ}}\frac{1}{2}\sum_{n}[(\bm{F}_{pQ}X_{Q}^{n}+u^{0}_{p})-x_{p}^{n}][(\bm{F}_{pR}X_{R}^{n}+u^{0}_{p})-x_{p}^{n}]$
(13)
$\displaystyle=\sum_{n}[(\bm{F}_{pQ}X_{Q}^{n}+u^{0}_{p})-x_{p}^{n}]\frac{\partial}{\partial
F_{iJ}}[(\bm{F}_{pR}X_{R}^{n}+u^{0}_{p})-x_{p}^{n}]$ (14)
$\displaystyle=\sum_{n}[(\bm{F}_{pQ}X_{Q}^{n}+u^{0}_{p})-x_{p}^{n}]\,[\delta_{ip}\delta_{JR}X_{R}^{n}]$
(15)
$\displaystyle=\sum_{n}[\bm{F}_{iQ}X_{Q}^{n}+u^{0}_{i}-x_{i}^{n}]\,[X_{J}^{n}],$
(16)

which, when indices are renamed and the expression is rearranged, produces

$\displaystyle\bm{F}^{GB}_{iJ}=\Big{(}\sum_{m}(x_{i}^{m}-u^{0}_{i})\,X_{Q}^{m}\Big{)}\Big{(}\sum_{n}\,X_{Q}^{n}\,X_{J}^{n}\Big{)}^{-1}$
(17)

Repeating the process for $\bm{u}$:

$\displaystyle 0$ $\displaystyle\overset{!}{=}\frac{\partial}{\partial
u^{0}_{i}}\frac{1}{2}\sum_{n}[(\bm{F}_{pQ}X_{Q}^{n}+u^{0}_{p})-x_{p}^{n}][(\bm{F}_{pR}X_{R}^{n}+u^{0}_{p})-x_{p}^{n}]$
(18)
$\displaystyle=\sum_{n}[(\bm{F}_{pQ}X_{Q}^{n}+u^{0}_{p})-x_{p}^{n}]\frac{\partial}{\partial
u^{0}_{i}}[(\bm{F}_{pR}X_{R}^{n}+u^{0}_{p})-x_{p}^{n}]$ (19)
$\displaystyle=\sum_{n}[(\bm{F}_{pQ}X_{Q}^{n}+u^{0}_{p})-x_{p}^{n}][\delta_{ip}]$
(20) $\displaystyle=\sum_{n}[(\bm{F}_{iQ}X_{Q}^{n}+u^{0}_{i})-x_{i}^{n}]$ (21)

Rearranging again produces the result

$\displaystyle u^{0}_{i}=\frac{1}{N}\sum_{n}(x_{i}^{n}-\bm{F}_{iQ}X_{Q}^{n})$
(22)

The stationarity conditions for $\bm{F}^{GB}$ and $\bm{u}^{0}$ can be solved
iteratively, or can be combined into a single expression depending on the size
of the problem. ∎