$\displaystyle=2\Big{[}(J_{2}^{-1}J_{1}-I)\mathbf{y}\cdot\eta+(\mathbf{y}-\mathbf{x})\cdot\eta\Big{]}$
$\displaystyle=2(\mathbf{y}-\mathbf{x})\cdot\eta-2h\sum_{i=2}^{n}y_{i}\cdot\eta_{i-1}+O(h^{2})\|\mathbf{y}\|.$

For Assumption 2.8, define $\hat{\mathbf{x}}$ as
$\hat{\mathbf{x}}_{i}:=\sum_{j=i}^{n}\frac{t^{j-i}}{(j-i)!}\mathbf{x}_{j}$ for
$i=1,\dots,n$, and consider the change of variable
$\mathcal{T}_{h}(\mathbf{x})=\hat{\mathbf{x}}$. Assumption 2.8 holds with this
change of variable and, indeed, one can easily check that

$\displaystyle
c_{h}(\mathbf{x},\mathcal{T}_{h}(\mathbf{x})+\sigma\mathbf{z})\leq$
$\displaystyle
Ch^{2-2n}\sigma^{2}\|\mathbf{z}\|^{2},\qquad\textrm{and}\qquad|f(\mathcal{T}_{h}(\mathbf{x})+\sigma\mathbf{z})-f(\mathbf{x})|\leq
C\|\sigma z_{n}\|.$

∎

## Acknowledgement

We would like to thank the anonymous referees for their useful suggestions for
the improvement of the paper. D.A was supported by The Maxwell Institute
Graduate School in Analysis and its Applications, a Centre for Doctoral
Training funded by the UK Engineering and Physical Sciences Research Council
(grant EP/L016508/01), the Scottish Funding Council, Heriot-Watt University
and the University of Edinburgh. M. H. Duong was supported by EPSRC Grants
EP/W008041/1 and EP/V038516/1. G.d.R. acknowledges support from the _Fundaç
$\tilde{\text{a}}$o para a Ci$\hat{e}$ncia e a Tecnologia_ (Portuguese
Foundation for Science and Technology) through the project UIDB/00297/2020
(Centro de Matemática e Aplicaç$\tilde{\text{o}}$es CMA/FCT/UNL).

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††thanks: L.F. and T.B. contributed equally to this work.††thanks: L.F. and
T.B. contributed equally to this work.

# Dual phase patterning during a congruent grain boundary phase transition in
elemental copper

Lena Frommeyer  Max-Planck-Institut für Eisenforschung GmbH, Max-Planck-Straße
1, 40237 Düsseldorf, Germany  Tobias Brink<EMAIL_ADDRESS>Max-Planck-
Institut für Eisenforschung GmbH, Max-Planck-Straße 1, 40237 Düsseldorf,
Germany  Rodrigo Freitas  Department of Materials Science and Engineering,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Timofey
Frolov<EMAIL_ADDRESS>Lawrence Livermore National Laboratory, Livermore, CA
94550, USA  Gerhard Dehm<EMAIL_ADDRESS>Max-Planck-Institut für Eisenforschung
GmbH, Max-Planck-Straße 1, 40237 Düsseldorf, Germany  Christian H. Liebscher
<EMAIL_ADDRESS>Max-Planck-Institut für Eisenforschung GmbH, Max-Planck-
Straße 1, 40237 Düsseldorf, Germany

(28th August 2024)

###### Abstract

The phase behavior of grain boundaries can have a strong influence on
interfacial properties. Little is known about the emergence of grain boundary
phases in elemental metal systems and how they transform. Here, we observe the
nanoscale patterning of a grain boundary by two alternating grain boundary
phases with distinct atomic structures in elemental copper by atomic
resolution imaging. The same grain boundary phases are found by grain boundary
structure search indicating a first-order transformation. Finite temperature
atomistic simulations reveal a congruent, diffusionless transition between
these phases under ambient pressure. The patterning of the grain boundary at
room temperature is dominated by the grain boundary phase junctions separating
the phase segments. Our analysis suggests that the reduced mobility of the
phase junctions at low temperatures kinetically limits the transformation, but
repulsive elastic interactions between them and disconnections could
additionally stabilize the pattern formation.

##  I Introduction

Grain boundaries (GBs) are the interfaces separating adjoining crystallites
and can impact the mechanical [1, 2, 3, 4] and electronic [5, 6, 7, 8]
properties of polycrystalline materials. GBs can exist in multiple stable and
metastable states, which are typically associated with differences in the
atomic structure of the GB core, and it was proposed that they can undergo
phase transitions [9, 10, 11, 12, 13, 14]. The terms “GB phase” [13] or
“complexion” [11, 12, 14] have been introduced as analogs to bulk phases to
underline that these interface phases can only exist in contact to other bulk
phases. Each GB phase is characterized by distinct thermodynamic excess
properties [9, 10, 15], which can have an impact on, for example, sliding
resistance [16], GB migration [17], or shear-coupled GB motion [18]. In
metallic systems, most experimental evidence for GB phase transitions is
inferred indirectly from abrupt changes in diffusivity [19, 20, 21] or GB
migration [22]. One of the first direct observations of two different
structures within one GB was obtained for NiO [23].

Experimental evidence of congruent GB phase transitions in elemental metals is
lacking, since they are difficult to observe. A congruent GB phase transition
is characterized by transformations limited to the GB core without a change in
grain misorientation and GB plane [12]. These transitions have mostly been
studied using atomistic modelling of [001] tilt GBs in fcc copper [15, 24,
25], in various tungsten GBs [26], and magnesium [27]. Recently, two different
GB phases were observed experimentally in $\Sigma$19b $\langle 1\,1\,1\rangle$
$\\{1\,7\,8\\}$ GBs in copper by atomic resolution scanning transmission
electron microscopy (STEM) [28]. Using atomistic simulations, it was found
that only one GB phase was stable over the temperature range from
$0\text{\,}\mathrm{K}$ to $800\text{\,}\mathrm{K}$ at ambient pressure and a
congruent phase transition would only be possible by applying tensile or shear
stresses [28]. The room temperature observations were related to stresses
stabilizing the metastable GB phase and a reduced mobility of the GB phase
junction at low temperatures, kinetically trapping the high temperature phase.

Grain boundary phase junctions themselves, which are line defects separating
two GB phases [13], therefore play an important role in the energetics and
kinetics of GB phase transitions. The junctions have a dislocation character,
similar to disconnections [29, 30], but their Burgers vectors include
contributions from the structural difference of the abutting GB phases [31].
Since they interact elastically via stress fields in a similar manner as
dislocations do [30], it is expected that they significantly contribute to the
nucleation barrier of GB phases and interact with other GB defects [31].
However, their character and related influence on GB phase transitions has
barely been studied [32].

In the present work, we investigate diffusionless, congruent GB phase
transitions in $\Sigma$37c $\langle 1\,1\,1\rangle$ $\\{1\,10\,11\\}$ GBs by
atomic resolution STEM and atomistic modelling. We find that two GB phases can
transform into each other at ambient pressure by temperature alone. We further
discuss the structure and properties of the different GB phases as well as
their thermodynamic stability. The influence of GB defects and phase junctions
on an experimentally observed GB phase pattering is discussed.

##  II Results

Figure 1: HAADF-STEM images of two different positions in a nearly symmetric
$\Sigma$37c $\langle 111\rangle$ $\\{1\leavevmode\nobreak\
10\leavevmode\nobreak\ 11\\}$ GB, in which the misorientation between both
grains is $None$. Even though the GB plane remains unchanged, two different
phases are visible, separated by a phase junction. The scale bars represent
$2\text{\,}\mathrm{n}\mathrm{m}$. Figure 2: Overviews of two more than
$300\text{\,}\mathrm{n}\mathrm{m}$ long GB segments assembled from multiple
HAADF-STEM images of near-symmetric areas of a $\Sigma 37\mathrm{c}$ $\langle
111\rangle$ GB. (a)–(b) The GBs consist of multiple, alternating domino (red)
and pearl (blue) segments. The lengths of each segment are indicated as well
as the deviation off the symmetric GB plane (green line). (c) Magnified view
of the orange region in (a). The domino segments in symmetric areas (less than
$None$ off the symmetric case) are between 40 and
$60\text{\,}\mathrm{n}\mathrm{m}$ long, whereas the pearl segments are shorter
with $10\text{\,}\mathrm{n}\mathrm{m}$ to $40\text{\,}\mathrm{n}\mathrm{m}$. A
high-quality copy of this image can be found in the Supplementary Material.

###  II.1 Experimental observation of GB phases

A $1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ thick $\langle 111\rangle$
epitaxially grown Cu thin film was used as a template material (see
Supplementary Fig. Dual phase patterning during a congruent grain boundary
phase transition in elemental copper for an inverse pole figure map of the
film). Plane-view focused ion beam (FIB) lamellas of $\Sigma$37c $\langle
111\rangle$ GBs were lifted out and their atomic structures were investigated
by high-angle annular dark-field (HAADF) STEM . Two different GB structures
could be observed, both of them occurring frequently. Fig. 1 shows the
structures and the phase junctions between them at two different positions
along a nearly symmetric GB. We termed these distinct GB phases pearl (blue)
and domino (red) due to their similarity to the structures in $\Sigma$19b
$\langle 111\rangle$ $\\{1\leavevmode\nobreak\ 7\leavevmode\nobreak\ 8\\}$ GBs
[28]. The misorientation between both grains is $None$, which is within the
Brandon criterion [33] for a $\Sigma$37c GB (nominal misorientation angle of
$None$).

Two different overview montages consisting of multiple individual HAADF-STEM
images encompassing a total length of up to $300\text{\,}\mathrm{n}\mathrm{m}$
of the GB are shown in Fig. 2. The GB adopts a slight curvature and deviates
in some regions from the symmetric orientation. Interestingly, the GB is
composed of alternating pearl and domino segments. Taking into account only
fragments deviating less than $None$ from the symmetric case, the domino GB
phase comprises 77$\%$ of the GB with segment lengths ranging from
$40\text{\,}\mathrm{n}\mathrm{m}$ to more than
$100\text{\,}\mathrm{n}\mathrm{m}$. The pearl GB phase segments adopt lengths
between $10\text{\,}\mathrm{n}\mathrm{m}$ to
$25\text{\,}\mathrm{n}\mathrm{m}$, taking up a total fraction of about 23$\%$.
These observations suggest that the domino phase is more stable at low
temperatures, but the large amount of remaining pearl phase is surprising: At
constant stress, the phase coexistence region for congruent GB phase
transitions of elemental systems is restricted to a single temperature,
thereby practically excluding thermodynamically stable coexistence [14]. To
understand the observed GB phase patterning by two structurally distinct GB
phases, we first explore their thermodynamic excess properties and defects in
detail. This includes the phase junction and the strain field of possible
disconnections compensating the slightly asymmetric orientation of the GB
segments. In addition, the influence of the phase junction kinetics is
considered.

###  II.2 Structure and properties of the grain boundary phases

Atomic resolution HAADF-STEM images of both pearl and domino GB phases are
shown in Fig. 3a and c. Furthermore, both grain boundary structures shown in
Fig. 3b and d were obtained by molecular statics simulations at
$0\text{\,}\mathrm{K}$ using an embedded-atom method (EAM) potential [34].
Here, two fcc half-crystals were joined and different relative displacements
were sampled following the $\gamma$-surface method until the experimentally-
observed structures were obtained.

Figure 3: (a) Domino and (c) pearl phase in a symmetric $\Sigma$37c $\langle
111\rangle$ $\\{1\leavevmode\nobreak\ 10\leavevmode\nobreak\ 11\\}$ GB in Cu
observed by HAADF-STEM. The overall misorientation between both grains is
about $None$ and the sub-atomic structures are indicated. The scale bars
represent $2\text{\,}\mathrm{n}\mathrm{m}$. (b),(d) Corresponding structures
obtained from atomistic simulations with the EAM potential by Mishin et al.
[34]. The same motifs as in the experiment are obtained. The black lines
indicate the unit cell of the GB structure.

The domino phase consists of domino I and II motifs, which can be mapped onto
each other by a $None$ rotation around the $\langle 3\leavevmode\nobreak\
4\leavevmode\nobreak\ 7\rangle$ direction parallel to the GB. In structural
unit notation [35], we denote the domino unit cell as $|$D  D $|$.
Disconnections take the form of extending or shortening one of the D motifs by
a square or change the arrangement of both D motifs as highlighted in
Supplementary Fig. Dual phase patterning during a congruent grain boundary
phase transition in elemental copper. The pearl phase is more complex and
consists of square motifs (S), either connected by a pearl chain (P) or a “B”
unit after two or three repetitions. The structural unit obtained by the
$\gamma$-surface method (Fig. 3d) has a $|$S P  S B P$|$ unit cell, indicating
that the varying distance between B units in the experiment represents
defects, which possibly compensate the slight asymmetry of the GB and slightly
smaller misorientation angle between both grains ($None$ instead of the
perfect $None$).

Figure 4: Grain boundary phases predicted by the EAM potential discovered with phase-space sampling by an evolutionary algorithm. (a)–(c) Pair plots of different GB excess properties that make the separation into two main clusters of data points visible. The color coding is according to a clustering algorithm that takes several excess properties into account. Many of the structures that were discovered are simply defective, i.e., they contain disconnections or other defects. Three defect-free base structures are highlighted by the triangle, diamond, and square symbols, in which two are microstates of the pearl phase (the triangle and diamond) and one refers to the domino phase (the square). The best predictor to separate the clusters is $[\tau_{22}]$, but the excess shear also provides a good indicator. (d) When plotting the GB energy over this predictor, a clear separation between low-energy domino-like and pearl-like structures can be seen. The low-energy pearl variants consist of either pearl #1 or #2, or a mixture of the two (indicated by the line connecting the data points and explored in more detail in Supplementary Fig. Dual phase patterning during a congruent grain boundary phase transition in elemental copper). (e)–(g) Snapshots of the low-energy structures from two directions. The unit cells are marked by black lines. | domino | pearl #1 | pearl #2 |
---|---|---|---|---
$\gamma_{0}$ | $0.857$ | $0.868$ | $0.874$ | J/m2
$[V]$ | $0.200$ | $0.170$ | $0.160$ | Å
$[\tau_{11}]$ | $-0.18$ | $-0.20$ | $-0.20$ | J/m2
$[\tau_{22}]$ | $\phantom{+}0.24$ | $-0.71$ | $-1.07$ | J/m2
$[\tau_{12}]$ | $\phantom{+}0.0\phantom{0}$ | $\pm 0.25$ | $\pm 0.07$ | J/m2
$[B_{1}]$ | $0.359$ | $\phantom{+}1.034$ | $\phantom{+}1.028$ | Å
$[B_{2}]$ | $0.000$ | $\pm 0.105$ | $\pm 0.210$ | Å
$[n]$ | $0$ | $0$ | $0$ |
Table 1: Excess properties of the three low-energy GB structures in the
$\Sigma$37c $\langle 111\rangle$ $\\{1\,10\,11\\}$ GB as predicted by the
computer model. We follow the conventions used by Frolov and Mishin [36],
where index 1 corresponds to the tilt axis $\langle 1\,1\,1\rangle$, 2
corresponds to the $\langle 3\,4\,7\rangle$ direction parallel to the GB, and
3 corresponds to the grain boundary normal $\langle 1\,10\,11\rangle$. The
signs of $[\tau_{12}]$ and $[B]$ depend on the choice of a specific
orientation of the phase and the coordinate system (see Supplementary Fig.
Dual phase patterning during a congruent grain boundary phase transition in
elemental copper for details). $[B_{3}]$ is equal to $[V]$ when $[n]=0$.

In addition to matching the experimental structures, we also sampled the phase
space of possible GB structures at $T=$0\text{\,}\mathrm{K}$$ efficiently—even
for structures that have an excess number of atoms per unit cell—using an
evolutionary algorithm [25] implemented in the USPEX code [37, 38]. The
thermodynamic excess properties of all structures are shown in Fig. 4a–d and
Supplementary Fig. Dual phase patterning during a congruent grain boundary
phase transition in elemental copper. These include the grain boundary energy
$\gamma_{0}$ at $0\text{\,}\mathrm{K}$, the excess volume $[V]$, the excess
stresses $[\tau_{ij}]$, the excess number of atoms $[n]$ (see Methods for its
definition), and the excess shear $[\mathbf{B}]$. The latter corresponds to
the microscopic translation vector between the two crystallites when no
external stress is applied to the system. The notation $[Z]$ refers to the
excess of property $Z$ in a system with a grain boundary over a perfect
crystal with the same number of atoms [36]. In order to obtain intensive
values for all excess properties, $\gamma$, $[V]$, and $[\tau_{ij}]$ are
normalized by the grain boundary area.

Using $k$-means clustering with $k=2$ on the full dataset of all structures
with $\gamma_{0}<$0.95\text{\,}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$$, we can
cleanly separate the structures into pearl-like and domino-like phases and
their defective variants. This can be visualized in pair plots, where two
different properties are plotted against each other (Fig. 4a–d and
Supplementary Fig. Dual phase patterning during a congruent grain boundary
phase transition in elemental copper). The excess stress component
$[\tau_{22}]$ is the best single predictor separating the two phases. The
domino structure has the lowest grain boundary energy and represents the
ground state at $0\text{\,}\mathrm{K}$. The lowest-energy phase in the pearl
cluster, indicated with an orange diamond symbol in Fig. 4 and termed pearl
#1, does not resemble the experimentally observed one. The “B” motif is
replaced by an “$\Omega$” motif: $|$P S P  S $\Omega|$. Further manual search
revealed a pearl #2 variant with
$6\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{m}^{2}$ higher grain boundary
energy and the experimentally-observed B motif (blue triangle in Fig. 4). The
excess shears $[\mathbf{B}]$ differ by
$(0.006,0.105,0.131)\,$\mathrm{\SIUnitSymbolAngstrom}$$ between the two
variants and the other excess properties are similarly close. The properties
of the defect-free domino, pearl #1, and pearl #2 structures are listed in
Table 1. There are several intermediary structures in between pearl #1 and #2
(indicated by the connecting line in Fig. 4d and explored in detail in
Supplementary Fig. Dual phase patterning during a congruent grain boundary
phase transition in elemental copper), which suggests that these states
represent different microstates of the pearl phase at elevated temperatures.
From STEM image simulations on pure pearl #1, pearl #2, and a mixture of both
stacked in $\langle 111\rangle$ direction (see Supplementary Fig. Dual phase
patterning during a congruent grain boundary phase transition in elemental
copper), we can conclude that the mixture would not be distinguishable from a
pure pearl #2 structure in experimental HAADF-STEM images.

A change in the excess number of atoms $[n]$ has been connected to diffusion-
driven grain-boundary phase transitions in various metals [15, 25, 39, 26].
However, in the present case all ground state structures of the defect-free GB
phases adopt values of $[n]=0$ (see Table 1), indicating that the GB phase
transition is not driven by the insertion or removal of atoms and is thus
considered to be diffusionless. Here, interstitial- or vacancy-type defects do
not lead to different grain boundary phases, but only to defective microstates
(Supplementary Figs. Dual phase patterning during a congruent grain boundary
phase transition in elemental copperc and Dual phase patterning during a
congruent grain boundary phase transition in elemental copper).

###  II.3 Diffusionless grain boundary phase transition

Figure 5: Free energy calculation with the EAM potential using the quasi-
harmonic approximation. (a) Plot of the GB free energies $\gamma$ for the
three low-energy structures we discovered. The dashed lines indicate an
approximation with quantum-mechanical effects, the solid lines represent a
purely classical approximation. The classical approximation yields equivalent
results above $100\text{\,}\mathrm{K}$. (b) The free energy differences show
that the domino phase is stable below approximately $460\text{\,}\mathrm{K}$,
while pearl #2 is stable above that temperature.

In a first step to explore the underlying mechanisms leading to the
experimentally observed patterning of the GB phases, we calculated their
excess free energies to determine GB phase stability and phase transition
temperatures. We used the quasi-harmonic approximation [40, 41] on the defect-
free structures (Fig. 5) and confirmed the results using thermodynamic
integration [42, 41] (see Methods and Supplementary Fig. Dual phase patterning
during a congruent grain boundary phase transition in elemental copper). Fig.
5 shows that this system exhibits a GB phase transition temperature of around
$$460\text{\,}\mathrm{K}$=0.33T_{m}$ under constant, ambient pressure, with
$T_{m}=$1358\text{\,}\mathrm{K}$$ being the melting point of copper. Domino is
the stable GB phase at low temperature, and pearl #2 at high temperature.

Figure 6: Phase transition simulations. (a) We start from the domino structure
with periodic boundary conditions (PBC) in the GB plane and open boundaries
normal to it. (b) The pearl phase nucleates at $800\text{\,}\mathrm{K}$ and
(c) keeps growing after reducing the temperature to $550\text{\,}\mathrm{K}$
(snapshot after $4\text{\,}\mathrm{n}\mathrm{s}$). The pearl structures
contain many defects (disconnections, black). For a slowly cooled pearl sample
with less defects see Supplementary Fig. Dual phase patterning during a
congruent grain boundary phase transition in elemental copper. (d) Regrowth of
the domino phase at $300\text{\,}\mathrm{K}$ observed after
$2\text{\,}\mathrm{n}\mathrm{s}$ for a system thickness of 3 atomic layers in
tilt direction. All pearl phase disappeared after an additional
$0.2\text{\,}\mathrm{n}\mathrm{s}$. The initial state was extracted from (b).
All images are slices of width $1\text{\,}\mathrm{n}\mathrm{m}$.

We used annealing simulations to test the prediction of the phase transition
temperature and to obtain partially transitioned systems containing GB phase
junctions. The nucleation and phase transition is expected to be quite slow on
MD timescales, so we started by annealing a sample containing the domino phase
(Fig. 6a) at $800\text{\,}\mathrm{K}$. This sample had open surfaces in
$\langle 1\,10\,11\rangle$ direction, but was otherwise periodic so that the
grain boundary had no contact to the open surfaces and nucleation was
homogeneous. Its width in tilt axis direction was
$6.3\text{\,}\mathrm{n}\mathrm{m}$, corresponding to 30 $\\{111\\}$ layers.
After around $1.3\text{\,}\mathrm{n}\mathrm{s}$, the pearl phase began to
nucleate and grow from within the parent domino phase (Fig. 6b) indicating
that homogeneous nucleation is possible, likely because the phase transition
requires no diffusion. This fits to the experimental observation of multiple
nanoscale segments of the pearl phase occurring within the GB (Fig. 2),
instead of nucleation only at e.g. GB triple junctions. Thus, multiple
nucleation sites for the pearl phase seem to be easily accessible within the
domino structure. We took the sample from Fig. 6b and annealed it further at
$550\text{\,}\mathrm{K}$, which lead to a growth of the pearl phase segment,
since it is above the transition temperature of $460\text{\,}\mathrm{K}$ (Fig.
6c). Below the transition temperature, the migration of the GB phase junction
is too slow to be observed in MD timescales. We could accelerate the process
by reducing the system to a thickness of three atomic $\\{1\,1\,1\\}$ layers,
in which case the pearl phase was observed to dissolve at
$300\text{\,}\mathrm{K}$ (Fig. 6d). A similar dependence of phase junction
mobility on system thickness was observed before [28].

The high temperature pearl phase was observed to consist of a mixture of both
pearl #1 and #2 variants, but distinguishing them is difficult due to the
large amount of defects, especially in between the “S” motifs. A cleaner pearl
GB phase could be obtained by further annealing at $800\text{\,}\mathrm{K}$
and subsequent cooling (Supplementary Fig. Dual phase patterning during a
congruent grain boundary phase transition in elemental copper). This phase
contains both the B and $\Omega$ structural units of the two pearl variants
stacked in $\langle 111\rangle$ direction, indicating that pearl #1 and #2
resemble microstates of a combined pearl phase, at least at high temperatures
in the model.

Finally, we excluded that diffusion could lead to the appearance of other new
phases by annealing a pearl grain boundary at $800\text{\,}\mathrm{K}$ for
$30\text{\,}\mathrm{n}\mathrm{s}$ with surfaces in $\langle 347\rangle$
direction. It has been shown in previous work [15] that such boundary
conditions are conducive to GB phase transitions, but no novel phase appeared
in our simulations, supporting the conclusion that pearl is the stable phase
at high temperatures.

###  II.4 Grain boundary phase junction

Figure 7: Burgers vector of the phase junctions. (a)–(b) Microscopic
translations $[\mathbf{B}]$ between the crystallites for the different phases.
The differences in $[\mathbf{B}]$ correspond to the Burgers vector
$\textbf{b}_{\text{junction}}$. Different possible $[\textbf{B}]$ vectors are
equivalent, since they are connected by DSC vectors $d^{\text{SC}}$. (c)
Burgers circuit on an actual junction in the simulation. The black lines were
chosen to be parallel, so that
$\overline{\mathrm{AB}}+\overline{\mathrm{CD}}=\mathbf{0}$. The green lines of
the circuit were measured in ideal unit cells of the GB phases to avoid
elastic distortions near the junction. (d) Burgers circuit on the experimental
image from Fig. 1a using the same method. (e)–(f) The lines crossing the GB
were translated to reference segments far away from phase junctions before
measuring in order to reduce elastic distortions. The same images as in Fig. 3
were used. The components parallel and normal to the GB were measured to be
$b_{\parallel}=-0.28$ to $-0.45\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ and
$b_{\perp}=-0.1$ to $+$0.14\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$. For more
circuits at different phase junctions see Supplementary Fig. Dual phase
patterning during a congruent grain boundary phase transition in elemental
copper.

The GB phase junction is a 1D line defect that separates both the pearl and
domino GB phases [13]. Besides being important for the kinetics of the GB
phase transition, it plays a vital role in the nucleation of GB phases and the
coalescence of GB phase segments. It was recently established that the phase
junction is characterized by a Burgers vector, which depends on the excess
properties of the abutting GB phases [31]. In a single GB phase, line defects
have disconnection character and their Burgers vectors are also displacement
shift complete (DSC) vectors [30]. This is not necessarily the case for phase
junctions, where the smallest possible Burgers vector is the difference in the
translation vector $[\mathbf{B}]$ of the abutting GB phases [31].
Nevertheless, the phase junction can also absorb disconnections and we
therefore started by finding the dichromatic pattern and DSC vectors of the
$\Sigma$37c GB (Supplementary Figs. Dual phase patterning during a congruent
grain boundary phase transition in elemental copper and Dual phase patterning
during a congruent grain boundary phase transition in elemental copper). We
then explored the possible $[\mathbf{B}]$ vectors of the GB phases by
constructing bicrystals with different values of $[B_{1}]$ and $[B_{2}]$ and
running molecular statics ($[B_{3}]$ is equal to the grain boundary excess
volume for $[n]=0$ and does not require systematic search). We found that
apart from the values listed in Table 1, $[B_{2}]$ can take any value obtained
by adding an integer multiple of, e.g., the DSC vector
$\mathbf{d}^{\text{SC}}_{1}=(0,a/74\cdot|347|,0)\approx(0,$0.420\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,0)$,
as expected (Fig. 7a–b).

The resulting Burgers vector of the phase junctions separating the GB phases
without additional disconnection content is shown in Fig. 7a–b and has a value
of

$\displaystyle\mathbf{b}_{\text{junction}}$
$\displaystyle=\mathbf{B}_{\text{pearl}}-\mathbf{B}_{\text{domino}}$
$\displaystyle=($0.669\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,\pm\frac{a}{148}|347|,$-0.04\text{\,}\mathrm{\SIUnitSymbolAngstrom}$)$
$\displaystyle\approx(0.669,\pm
0.210,-0.04)\,$\mathrm{\SIUnitSymbolAngstrom}$.$ (1)

In addition, we cooled the simulation from Fig. 6c to $200\text{\,}\mathrm{K}$
and minimized the structure with regard to the potential energy. We then
constructed a Burgers circuit (Fig. 7c) around one of the junctions. The
circuit was chosen to contain two parallel, equally long lines (black) that
add to zero. The Burgers vector can then be calculated as
$\mathbf{b}_{\text{junction}}=\overline{\mathrm{BC}}+\overline{\mathrm{DA}}$,
where the two lines across the GB are measured in single-phase simulation
cells to avoid elastic distortion due to the junction and the defects in the
pearl phase (see Methods for details on the procedure [31]). We obtained a
value of $(0.672,-0.205,-0.039)\,$\mathrm{\SIUnitSymbolAngstrom}$$, which
corresponds to the predicted value within the expected accuracy of the atomic
positions in the simulation.

We investigated the Burgers vector of the phase junction experimentally using
the same method [31]. A complete Burgers circuit is drawn clockwise around the
phase junction as shown in Fig. 7d. The lines crossing the GB were measured in
reference images far from the junction. As we observe a projection of the
phase junction by HAADF-STEM, only the second and third component, parallel
and normal to the GB plane, can be deduced. We calculated a Burgers vector for
in total three different phase junctions (see Fig. 7 and Supplementary Fig.
Dual phase patterning during a congruent grain boundary phase transition in
elemental copper). The values of the Burgers vector at the phase junction
shown in Fig. 7d ($\mathbf{b}_{a}$) and the ones shown in Supplementary Fig.
Dual phase patterning during a congruent grain boundary phase transition in
elemental coppera ($\mathbf{b}_{b}$) and b ($\mathbf{b}_{c}$) have values of

$\displaystyle\mathbf{b}_{a}$ $\displaystyle=(?,-0.28\text{ to
}$-0.45\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,+0.14\text{ to
}$-0.1\text{\,}\mathrm{\SIUnitSymbolAngstrom}$)$ (2)
$\displaystyle\mathbf{b}_{b}$ $\displaystyle=(?,+0.33\text{ to
}$+0.48\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,+0.05\text{ to
}$+0.2\text{\,}\mathrm{\SIUnitSymbolAngstrom}$)$ (3)
$\displaystyle\mathbf{b}_{c}$ $\displaystyle=(?,+0.25\text{ to
}$+0.38\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,+0.05\text{ to
}$+0.26\text{\,}\mathrm{\SIUnitSymbolAngstrom}$)$ (4)

depending on the Burgers circuit drawn around the phase junction. The
uncertainty of each measurement is about $\pm
0.2\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, considering possible sources of
error in the experiment, such as small localised residual stresses in the
undeformed reference state and non-linear scan distortions leading to sub-
Ångström variations in the positions of atomic columns. Thus, the values are
in good agreement with $\mathbf{b}_{\text{junction}}$ obtained from the
computer model. The experimental determination of the Burgers vector of the GB
phase junction may be further complicated by additional disconnections next to
or within the phase junction. The smallest possible disconnections in a
$\Sigma$37c system are shown in Supplementary Fig. Dual phase patterning
during a congruent grain boundary phase transition in elemental coppera,
having a minimum length of $0.21\text{\,}\mathrm{\SIUnitSymbolAngstrom}$
parallel to the GB, the $b_{2}$ component, and
$0.12\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ normal to the GB, the $b_{3}$
component. These smallest disconnections could be added to or subtracted from
the experimentally obtained values of the phase junctions’ Burgers vector and
the value would still be within the range determined from atomistic
simulations.

The component of the Burgers vector along the tilt axis of the GB and hence
the line sense of the phase junction,
$b_{1}=$0.67\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$, is the largest
component. Thus the phase junction predominantly adopts a screw-type
character. This is consistent with the observations for $\Sigma 19$b GBs [28],
where the phase junction between both GB phases was investigated qualitatively
without further calculations. The absolute quantitative evaluation of the
Burgers vector is needed to calculate the elastic interactions between GB
phase junctions and disconnections. As we will see in the next section, this
elastic interaction is mainly responsible for the patterning of both GB phases
observed in our experiments at room temperature.

###  II.5 Mechanisms leading to grain boundary phase patterning

We now discuss the mechanisms leading to the experimentally observed
patterning of the GB by both pearl and domino phases. When the sample is
cooled down to room temperature from above $460\text{\,}\mathrm{K}$, the phase
transition initiates by nucleation of the domino phase within the pearl phase.
Similar to bulk phase transitions, nuclei of the domino phase appear due to
random thermal fluctuations and will most likely start at the surfaces or
interfaces, which lower the energy needed to initiate the process. Detailed
investigations of a homogeneous nucleation of GB phases were recently
published by Winter et al [32]. The free energy change during the formation of
a nucleus can be written as the sum of the free energy reduction due to the
transition to the thermodynamically stable phase and the energy cost of the
phase boundary. The phase junction’s contribution consists of an elastic
interaction energy and a core energy. The core energy of a phase junction of a
$\Sigma 29$ $(520)$ $[001]$ symmetric tilt GB in tungsten was calculated to be
extremly anisotropic, being four times lower in the tilt direction compared to
the normal of the GB plane [32]. To our knowledge, this is the only reported
value of a GB phase junction core energy. However, in a $\Sigma 5\langle
100\rangle(210)$ GB in copper [13], a GB phase nucleus was observed to be
highly anisotropic as well, also showing an elongated shape in the tilt
direction. Thus we assume that the nucleus in the present case has an
approximately oval shape and rapidly expands along the $\langle 111\rangle$
tilt axis due to an anisotropic core energy. This implies that the nucleating
domino phase becomes multiple times longer in the $\langle 111\rangle$
direction than the $\langle 347\rangle$ one, matching the observed pattern
with segment lengths much shorter ($10$ to $100\text{\,}\mathrm{n}\mathrm{m}$)
than the film thickness ($1\text{\,}\mathrm{\SIUnitSymbolMicro m}$).

The free energy change due to a newly nucleated GB phase can then be
calculated by assuming it is enclosed by two parallel phase junction lines by
the expression

$\frac{\Delta G_{\text{nucl}}(T)}{t}=l\Delta\gamma(T)+E_{\text{dip}},$ (5)

where $l$ is the length of the newly formed phase segment, $t$ is the film
thickness, $\Delta\gamma(T)$ is the temperature dependent free energy
difference of the GB phases, and $E_{\text{dip}}$ is the energy of the phase-
junction dipole per unit line segment, which we assume to be temperature
independent, consisting of the core energy and the elastic interaction energy
$E_{\text{inter}}$. The elastic interaction energy was first described by
Nabarro for two dislocations [43] and later adapted for disconnections [30].
We assume that it is also valid for phase junctions, which have a dislocation
content [31]:

$\begin{split}\frac{E_{\text{inter}}}{t}=&-\frac{\mu}{2\pi}\ln\frac{l}{\delta_{0}}\left[(\mathbf{b}^{a}\cdot\hat{\mathbf{o}})(\mathbf{b}^{b}\cdot\hat{\mathbf{o}})+\frac{(\mathbf{b}^{a}\times\hat{\mathbf{o}})\cdot(\mathbf{b}^{b}\times\hat{\mathbf{o}})}{1-\nu}\right]\\\
&-\frac{\mu}{2\pi}\frac{\left(\mathbf{b}^{a}\cdot\hat{\mathbf{n}}\right)\left(\mathbf{b}^{b}\cdot\hat{\mathbf{n}}\right)}{1-\nu}.\end{split}$
(6)

Here, $\mathbf{b}^{a}$ and $\mathbf{b}^{b}$ are the Burgers vectors, $\mu$ is
the shear modulus
($\mu_{\text{Cu}}=$34\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$$),
$\hat{\mathbf{n}}$ the unit vector of the GB plane normal, $\hat{\mathbf{o}}$
the unit line vector (equal to the tilt axis in our case), $\nu$ the Poisson’s
ratio ($\nu_{\text{Cu}}\approx 0.34$), and $\delta_{0}$ the core size. We can
simplify the equation by choosing our coordinate system such that
$\hat{\mathbf{n}}=(0,0,1)$ and $\hat{\mathbf{o}}=(1,0,0)$:

$\begin{split}\frac{E_{\text{inter}}}{t}=&-\frac{\mu}{2\pi}\ln\frac{l}{\delta_{0}}\left[b^{a}_{1}b^{b}_{1}+\frac{b^{a}_{2}b^{b}_{2}+b^{a}_{3}b^{b}_{3}}{1-\nu}\right]\\\
&-\frac{\mu}{2\pi}\frac{b^{a}_{3}b^{b}_{3}}{1-\nu}.\end{split}$ (7)

Since the core size $\delta_{0}$ leads to a constant energy contribution
independent of the junction distance $l$, we can treat it as an effective core
size that already includes $2E_{\text{core}}$ and define
$E_{\text{dip}}=E_{\text{inter}}/t$ for simplification.

Once the oval nucleus expanded over the whole film thickness, the anisotropy
of the core energy can be neglected and the phase junctions can be simplified
by a pair of two dislocation lines having opposite Burgers vectors
($\mathbf{b}^{a}=-\mathbf{b}^{b}$) and thus an attractive interaction during
growth.

Figure 8: Free energy change due to a domino nucleus at
$T=$400\text{\,}\mathrm{K}$$. Calculated for a periodic system of alternating
pearl and domino phases with a period $L=$70\text{\,}\mathrm{n}\mathrm{m}$$.
Calculations for a system with only phase junction defects and for a system
with an additional disconnection per period. We assume
$\delta_{0}=$1\text{\,}\mathrm{n}\mathrm{m}$$ and subtract the value of
$\Delta G/t$ at $l_{\text{domino}}=$1\text{\,}\mathrm{n}\mathrm{m}$$, since
the offset of the curves depends on the exact value of the defect core
energies.

The growth of the domino phase inclusions can be limited by the migration of
the GB phase junction and the interaction of phase junctions when two
neighboring domains grow towards each other. The motion of the phase junction
is strongly temperature dependent and may contribute to a stagnation in growth
of the domino phase domains below a temperature of $400\text{\,}\mathrm{K}$,
as was also observed previously [28]. One reason for the reduced mobility
could be the large screw ($b_{1}$) component of the Burgers vector [44]. The
role of interacting phase junctions during GB phase coalescence is far less
understood. In a perfect GB, it is reasonable to assume that two newly
nucleated domino phases have the same translation vector $[\mathbf{B}]$ and
the two nuclei are thus delimited by junctions with the same Burgers vector
($\mathbf{b}_{\text{junction}}=\Delta[\mathbf{B}]$). This means that the two
closest junctions have opposite Burgers vectors and attract each other, which
would promote the coalescence of domino phase domains. It is conceivable that
the $[B_{2}]$ component of the two domino phases is different, but this does
not lead to repulsion of the junctions due to the large $b_{1}$ component of
their Burgers vectors. In this scenario, a patterning is not expected, which
is in line with the fact that it was also not observed in the simulations.

However, so far we did not consider the impact of additional disconnections in
the GB. They are most prominent in slightly asymmetric boundary segments
observed in the experiment, compensating for deviations in GB plane
inclination or a slight twist between both neighboring grains. One of the most
prominent disconnections in the experimental datasets is highlighted in
Supplementary Fig. Dual phase patterning during a congruent grain boundary
phase transition in elemental copper. Its Burgers vector was determined as
described for the phase junctions in the previous section to be
$\mathbf{b}_{\text{exp.
disc.}}=(?,0.23,-0.06)\,$\mathrm{\SIUnitSymbolAngstrom}$$ and its full Burgers
vector is thus close to the
$(2.09,0.21,-0.12)\,$\mathrm{\SIUnitSymbolAngstrom}$$ DSC vector as derived
from the dichromatic pattern (see Supplementary Fig. Dual phase patterning
during a congruent grain boundary phase transition in elemental copper). If
disconnections compensate the GB asymmetry or a twist between two grains,
their Burgers vectors must be equal (if they were opposite, the average GB
plane would remain symmetric or the twist would be undulating). The large
$b_{1}$ component cannot be explained by a compensation of a GB asymmetry
deviating from the symmetric $\\{1\,10\,11\\}$ GB plane alone, but implies
that a twist component along the $\langle 1\,1\,1\rangle$ axis is compensated
by such a disconnection. A disconnection occuring every
$70\text{\,}\mathrm{n}\mathrm{m}$ corresponds to a twist of about $None$. Such
a small twist component is likely to occur between two neighbouring grains due
to the unavoidable, slight roughness of the substrate. The electron
backscattered diffraction measurements showed deviations of up to $None$ from
the perfect $\langle 1\,1\,1\rangle$ orientation of different grains (see
Supplementary Fig. Dual phase patterning during a congruent grain boundary
phase transition in elemental copper), indicating that twist components are
likely to occur between two grains.

Fig. 8 shows the free energy of a domino nucleus $\Delta G_{\text{nucl}}/t$ in
a periodic pattern of alternating domino and pearl phases with a period of
$L=$70\text{\,}\mathrm{n}\mathrm{m}$$ (fitting the experimentally observed
period length and thus the average distance of the original nuclei). It was
calculated with a finite amount of periodic images, but converges quickly with
the number of images. The free energy is minimal when the whole pearl phase
disappears since the GB phase junctions have attractive interactions. We
introduced the experimentally observed disconnection, which is attracted by
one of the junctions and repulsed by the other, by assuming it merges with one
of the junctions into one combined Burgers vector. Here, a minimum appears at
40 to $50\text{\,}\mathrm{n}\mathrm{m}$ segment length for domino and 20 to
$30\text{\,}\mathrm{n}\mathrm{m}$ for pearl, corresponding to the experimental
patterning. An increase of the period length $L$ could be facilitated either
by disappearance of one or more pearl phase segments, which is connected to a
high energy barrier in the patterned GB, or by elongation of the total GB
length. In a real system, though, the GB length is typically fixed by triple
junctions and would have to increase its curvature. The pattern is thus
stabilised. It should also be noted that the relative regularity of the
observed pattern supports our defect interaction hypothesis. If the pearl
phase were left solely due to kinetic reasons, a more random arrangement would
be expected.

Figure 9: Schematic for the occurrence of the observed patterning of pearl and
domino phase. (f) Nearly-symmetric region from Fig. 2a shown for comparison.

The evolution of the patterning process is sketched in Fig. 9. At higher
temperatures, the grain boundary consists of pearl phase and possibly
disconnections to compensate for slight twist components of the GB. During
cooling, domino segments nucleate and disconnections are attracted to the
junctions. The growth of the domino phase is stopped by the repulsive
interaction between the combined junction/disconnection defect with the
undecorated junctions and a GB phase pattern appears.

##  III Conclusion

We observed two GB phases in a $\Sigma$37c $\langle 1\,1\,\overline{1}\rangle$
$\\{1\,10\,11\\}$ GB in a 1-$\mathrm{\SIUnitSymbolMicro m}$-thick elemental
copper film by HAADF-STEM. Over a more than $300\text{\,}\mathrm{n}\mathrm{m}$
long GB segment, these phases form an alternating pattern between segments of
domino phase ($40\text{\,}\mathrm{n}\mathrm{m}$ to
$100\text{\,}\mathrm{n}\mathrm{m}$ long segments) and pearl phase
($10\text{\,}\mathrm{n}\mathrm{m}$ to $25\text{\,}\mathrm{n}\mathrm{m}$ long
segments). Free energy calculations on the structures simulated with an EAM
potential show a diffusionless, congruent phase transition from domino (low
temperature) to pearl (high temperature) at around $460\text{\,}\mathrm{K}$.
In light of this, the observation of patterning at room temperature is
surprising, since the phase coexistence at ambient pressure is limited to a
single temperature by Gibb’s phase rule for elemental systems. Limited
kinetics of the GB phase junction motion, but also the elastic interaction
field of GB phase junctions and existing disconnections could play a
significant role. Therefore, we quantitatively determined the Burgers vectors
of the GB phase junctions and disconnections, which match the predictions from
the differences in excess shears $[\mathbf{B}]$ and DSC lattice, respectively.
By considering the elastic interactions between these defects, which resemble
those of lattice dislocations, we found that certain arrangements of defects
can energetically stabilize the phase pattern. While pure phase junctions
occur in pairs with opposite Burgers vectors and thus have attractive
interactions, which favour consolidation of a single phase in case of
sufficient mobility, the addition of regularly-spaced disconnections with the
same Burgers vectors can support the patterning. Such disconnections can occur
to compensate a slight twist component of the GB and are attracted to one half
of the phase junction pair and repulse the other. It is known that GB phase
transitions can influence material properties such as diffusion or GB mobility
and we therefore expect that phase patterning could open up new ways to
control such properties.

##  IV Methods

Specimen preparation
The TEM specimens have been extracted from a $\langle 111\rangle$ epitaxially
grown Cu thin film, deposited by molecular beam epitaxy on a $\langle
0001\rangle$ sapphire wafer at room temperature with post-deposition annealing
at $673\text{\,}\mathrm{K}$ for $3\text{\,}\mathrm{h}$. The films have been
deposited by the Central Scientific Facility Materials of the Max Planck
Institute for Intelligent Systems in Stuttgart.

Characterisation of the microstructure has been performed using a Thermo
Fisher Scientific Scios2HiVac dual-beam SEM equipped with an EBSD detector.
Two inverse pole figure maps of the electron backscatter diffraction scan are
shown in Supplementary Fig. Dual phase patterning during a congruent grain
boundary phase transition in elemental copper. Site-specific plane view FIB
lamellas have been lifted out and thinned using the Scios 2 DualBeam SEM/FIB
microscope, starting with a gallium ion beam voltage and current of
$30\text{\,}\mathrm{k}\mathrm{V}$ and $0.1\text{\,}\mathrm{n}\mathrm{A}$ and
ending at $5\text{\,}\mathrm{k}\mathrm{V}$, $16\text{\,}\mathrm{p}\mathrm{A}$.

Scanning transmission electron microscope imaging
The FIB lamellas were investigated with a probe-corrected FEI Titan Themis
60-300 (Thermo Fisher Scientific). The electrons, which are emitted by a high-
brightness field emission gun, were accelerated to
$300\text{\,}\mathrm{k}\mathrm{V}$. The probe current has been set to
70–$80\text{\,}\mathrm{p}\mathrm{A}$. The STEM datasets were registered with a
high-angle annular dark-field (HAADF) detector (Fishione Instruments Model
3000), using collection angles of
78–$200\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$ and a semi-
convergence angle of $17\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$.
The datasets consist of image series with 50–100 images and a dwell time of
$1\text{\,}\mathrm{\SIUnitSymbolMicro s}$. In order to reveal the structures,
as well as to reduce noise and instabilities of the instruments, these
datasets have been averaged and optimized by using a background substraction
filter, Butterworth filter and Gaussian filter, ensuring that the atomic
structure of the original image has been preserved.

MD simulations
MD simulations were performed using LAMMPS[45] (https://lammps.sandia.gov/)
with an EAM potential for Cu by Mishin et al. [34]. This potential has proven
to at least qualitatively capture the structures of $\langle 111\rangle$ tilt
grain boundaries before [28] and was developed with good agreement to the
phononic properties of copper [34], which is important for the free energy
calculations. All molecular dynamics simulation were performed with a time
integration step of $2\text{\,}\mathrm{f}\mathrm{s}$.

For a simple structure search using the $\gamma$-surface method, we
constructed a bicrystal out of two fcc crystallites with a size of
$31.1\text{\times}45.0\text{\times}6.26\text{\,}\mathrm{\SIUnitSymbolAngstrom}$
each (1312 atoms in total). The bottom crystal was oriented with
$[\overline{3}\,4\,7]$, $[11\,\overline{1}\,10]$, and $[1\,1\,\overline{1}]$
along the $x$, $y$, and $z$ directions. The top crystal orientation was
$[3\,4\,7]$, $[11\,\overline{10}\,1]$, and $[1\,1\,\overline{1}]$, resulting
in a misorientation angle of $None$. The $x$ and $z$ directions were periodic
and their length was kept fixed to preserve the ground state fcc lattice
constant of $a=$3.615\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$, while the $y$
direction contained open boundaries. We systematically displaced the top
crystal and minimized the energy of the system to sample those GB
configurations which do not require additional interstitial or vacancy atoms.

The lattice constant of fcc copper as a function of temperature was obtained
as described by Freitas et al. [42]. Annealing simulations were performed with
the same boundary conditions as the $0\text{\,}\mathrm{K}$ simulations above,
but with the lattice constant adjusted to the target temperature and with a
larger box size of
$630\text{\times}160\text{\times}63.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$
($526\,800$ atoms). One simulation with open boundaries in $x$ direction was
performed to investigate possible phase transitions requiring a particle
reservoir [15]. This simulation cell was chosen to be thicker in $y$ direction
and shorter in $x$ direction
($370\text{\times}230\text{\times}63.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$,
$435\,320$ atoms), to avoid any influence due to the diffusion-driven changes
on the surface during the long annealing time of
$30\text{\,}\mathrm{n}\mathrm{s}$.

All simulation results were visualized with OVITO [46].

Structure search and calculation of excess properties
Possible GB structures were sampled with an evolutionary algorithm [25]
implemented using the USPEX code [37, 38].

For these samples, excess properties were calculated in a region around the
grain boundary excluding atoms closer than $1\text{\,}\mathrm{n}\mathrm{m}$ to
the surface. We use the definition of the excess properties by Frolov and
Mishin [47, 36]. The excess number of atoms $[n]$ is defined in terms of a
fraction of a $\\{1\,10\,11\\}$ plane[15] as

$[n]=\frac{N}{N_{\\{1\,10\,11\\}}}\mod 1,$ (8)

where $N$ is the total number of atoms in the simulation cell and
$N_{\\{1\,10\,11\\}}$ corresponds to the number of atoms in a defect-free
$\\{1\,10\,11\\}$ plane of the fcc crystal.

The microscopic translation vector $[\textbf{B}]$ between the two crystallites
was computed by first constructing a dichromatic pattern in the first
crystallite far away from the grain boundary and extending this pattern to the
second crystallite. Now, $[B_{1}]$ and $[B_{2}]$ were obtained by shifting the
dichromatic pattern with fixed $[B_{3}]=[V]$ to fit the second crystallite. In
case of $[n]=0$ this is sufficient, but interstitial-like atoms or vacancy-
type defects in the grain boundary can also affect $[B_{3}]$ (see
Supplementary Fig. Dual phase patterning during a congruent grain boundary
phase transition in elemental copperd). For these cases, we also varied
$[B_{3}]$ to obtain a fit. The values of $[\mathbf{B}]$ were then restricted
to the DSC unit cell.

In order to separate the pearl and domino phases, we used a $k$-means
clustering algorithm as implemented in scikit-learn [48] on the $\gamma_{0}$,
$[V]$, $[\tau_{11}]$, $[\tau_{22}]$, $|[\tau_{12}]|$, $[n]$, and $[B_{1}]$
data. The $[B_{2}]$ data does not exhibit any pattern and was excluded. The
silhouette coefficient [49] indicates optimal clustering at $k=2$ clusters
(Supplementary Fig. Dual phase patterning during a congruent grain boundary
phase transition in elemental copperd)

STEM image simulation
STEM image simulations were performed using the multislice algorithm [50, 51].
An electron probe with $300\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$, semi-
angle of $17.8\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$, focal spread
of $100\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ and defocus of 0 was given to
match the settings used in the experiment. The HAADF-detector was set to
77.9–$200\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$. The step size was
selected as $0.178\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ to match to the
imaging conditions. A slice thickness of
$2\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ was used as the atomic column
separation in $z$-height is $2.09\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. All
simulated cells were of the same thickness of
$63\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ to ensure comparability.

Free energy calculation
Free energies were calculated using the quasi-harmonic approximation [40, 41].
Phononic eigenfrequencies were obtained from force constant matrices computed
with the dynamical_matrix command in LAMMPS. For each GB, a corresponding fcc
slab was produced with the same number of atoms and the same surfaces. This is
necessary, since the subsystem method described by Freitas et al.[41]
introduces an artificial boundary in the force constant calculation. The GB
free energy $\gamma(T)$ is then simply the free energy difference between
these two systems normalized to the grain boundary area. We confirmed the
results using thermodynamic integration [42] along the Frenkel–Ladd path [52]
with the subsystem method [41]. These results agree well except for the pearl
#1 structure (Supplementary Fig. Dual phase patterning during a congruent
grain boundary phase transition in elemental copper). The pearl #1 structure
started nucleating the B motif of the pearl #2 structure at temperatures above
$200\text{\,}\mathrm{K}$, leading to large dissipation during the
thermodynamic integration path. The data was discarded due to its
unreliability. This behavior of the pearl phase is nevertheless in accordance
with the expected phase stability predicted using the quasi-harmonic
approximation: The pure pearl #1 structure is less stable than pearl #2 (Fig.
5).

Burgers circuit
As described by Frolov et al. [31], a Burgers circuit is drawn around a phase
junction. The Burgers circuit is then split into 4 vectors: 2 vertical
vectors, crossing each the domino/pearl phase and 2 horizontal vectors,
described by specific planes in each grain next to the GB. The starting and
end points of the vectors across the GB phases are related to recognizable
features in the GB structures (green markers in Fig. 7c–e). The same vectors
are measured in regions far away from a phase junction, to aim for a stress-
free reference structure. Therefore, images taken in an area without phase
junction are used and the atomic positions of the same recognizable features
are localised by applying a Gaussian peak fitting algorithm [53]. They are
averaged over at least 4 identical sites in the reference states. Thereby,
differences up to $0.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ are observed,
which could be limited to an uncertainty of $\pm
0.1\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ since we averaged over several
measurements.

In the original description of the method [31], the horizontal lines in both
grains are parallel to the GB plane and cancel out each other. Here, this is
not the case as the planes needed to be shorter and still well defined. Since
these lines are completely in defect-free fcc regions, we can determine the
corresponding vector in the crystal coordinate system by counting atomic
columns along specific crystallographic directions. For the example shown in
Fig. 7d, these are $\mathbf{c}_{\mu}=a[34/6\leavevmode\nobreak\
46/6\leavevmode\nobreak\ 80/6]$ in the upper and
$\mathbf{c}_{\lambda}=a[34/6\leavevmode\nobreak\ -80/6\leavevmode\nobreak\
-46/6]$ in the lower grain. To add them up, we rotate the line in the lower
grain into the crystal coordinate system of the upper grain[54, 55] to obtain

$\displaystyle\mathbf{b}_{\mu}=\mathbf{c}_{\mu}+M_{1}\mathbf{c}_{\lambda},$
(9)

where $M_{1}$ is the corresponding rotation matrix. In a second step,
$\mathbf{b}_{\mu}$ needs to be separated into components normal and parallel
to the GB, so it is again multiplied by an appropriate rotation matrix
$M_{2}$. An error of $\pm 0.1\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ was
identified for these values taking into account the limitation of measuring
the misorientation between both grains and the GB plane ($\pm
0.3\text{\,}{}^{\circ}$). The Burgers vector results as the sum of all 4
vectors—the parts crossing each phase in a reference state as well as the
horizontal parts in both grains.

##  V Data availability

The main datasets of this study are published at
https://doi.org/10.5281/zenodo.5354071. Other data supporting the findings of
this study are available from the authors upon reasonable request.

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##  VI Acknowledgements

The authors thank Niels Cautaerts for helpful discussions and help with the
STEM image simulation. G. Richter and his team from the Max Planck Institute
for Intelligent Systems are gratefully acknowledged for producing the Cu thin
film by molecular beam epitaxy. This project has received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (Grant agreement No. 787446; GB-CORRELATE).
This work was partially performed under the auspices of the U.S. Department of
Energy (DOE) by the Lawrence Livermore National Laboratory (LLNL) under
Contract No. DE-AC52-07NA27344. T.F. acknowledges the funding by the
Laboratory Directed Research and Development Program at LLNL under Project
Tracking Code number 19-ERD-026.

##  VII Author contributions

T.B. and L.F. contributed equally to this work. L.F. performed the
experimental sample preparation, HAADF-STEM investigations and analysis of the
obtained datasets. C.H.L. and G.D. designed concept of the experimental study.
T.B. and R.F. calculated free energies using the quasi-harmonic approximation
and T.F. conducted the USPEX simulations. All other simulations and analyses
of the simulation data were performed by T.B. The project was supervised by
C.H.L. and G.D., who also contributed to discussions. G.D. secured funding for
L.F. and T.B. via the ERC grant GB-CORRELATE. L.F. and T.B. prepared the
initial draft and all authors contributed to the preparation of the final
manuscript.

##  VIII Competing interests

The authors declare no competing interests.

See pages 1 of supplementary-information.pdf See pages 2 of supplementary-
information.pdf See pages 3 of supplementary-information.pdf See pages 4 of
supplementary-information.pdf See pages 5 of supplementary-information.pdf See
pages 6 of supplementary-information.pdf See pages 7 of supplementary-
information.pdf See pages 8 of supplementary-information.pdf


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