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

# Atomic resolution observations of silver segregation in a [111] tilt grain
boundary in copper

Lena Langenohl  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  Gunther Richter  Max Plank Institute for Intelligent Systems,
Heisenbergstr. 3, 70569 Stuttgart, Germany  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

Alloying a material and hence segregating solutes to grain boundaries is one
way to tailor a material to the demands of its application. Direct observation
of solute segregation is necessary to understand how the interfacial
properties are altered. In this study, we investigate the atomic structure of
a high angle grain boundary both in pure copper and upon silver segregation by
aberration-corrected scanning transmission electron microscopy and
spectroscopy. We further correlate the experiments to atomistic simulations to
quantify the local solute excess and its impact on grain boundary properties.
We observe that the grain boundary structure remains intact upon silver
segregation and up to five different positions within a structural unit serve
as segregation sites. By combining the atomic resolution observation with
atomistic modelling, we are able to quantify the local silver concentration
and elucidate the underlying segregation mechanism.

##  I Introduction

Segregation of elements to grain boundaries (GBs) has a dramatic impact on
material properties. Examples include the material becoming stronger, or
generally having a more stable microstructure [1, 2, 3, 4, 5, 6, 7], but also
increasing the susceptibility to failure through corrosion or embrittlement
[8, 9, 10]. One of the most prominent examples of embrittlement is alloying
bismuth to copper, which was already discovered back in 1874 [11]. More recent
publications explain the embrittlement by a weakening of the metallic bonds at
the GB, which can be related to a pure atomic size effect or the change of the
electronic structure at the GB [12, 13, 14].

Early research on segregation revealed significant changes in segregation
behaviors for GBs with different misorientations between both grains using
autoradiographs, atomic absorption spectroscopy or Auger electron spectroscopy
[15, 16, 17, 18]. Even though such techniques can reveal chemical changes by
less than 1 at-$\%$ and the spatial resolution across the GB can go down to
$0.1\text{\,}\mathrm{n}\mathrm{m}$, the lateral resolution, revealing chemical
changes within the GB structure, is impossible. Furthermore, it was already
stated since a long time that also other GB parameters and thus the existing
GB and its atomic structure should play a crucial role, even though
experimental observations on the atomic scale were not available [19, 20, 21,
22, 23]. Segregation can change different properties of the existing GB phase,
such as its mobility, energy, stability, but also its atomic structure [24].
It can introduce mono- or bilayers [25, 12, 26] or precipitates of solute
atoms at the GB, and can even change the GB plane by faceting or induce a GB
phase transformation [27, 28, 29, 30]. Only few direct experimental
observations of segregation at the GB on the atomic level exist so far since
high-resolution imaging microscopes and spectroscopy are needed [4, 29, 30,
31, 32, 33, 34, 35, 12, 36]. To completely comprehend how segregation affects
macroscopic features, such as strengthening, embrittlement, or corrosion, we
must first identify the segregation-induced atomistic changes inside a GB
structure.

The fundamental thermodynamics implemented by Gibbs can be used to describe
adsorption at interfaces. Langmuir and McLean used these equations to develop
a model of GB segregation, describing the concentration limit of solutes at
the GB in dependence of the bulk concentration and temperature [37]. Their
model combines thermodynamics with statistical mechanics, using a finite
number of atomic positions in the GB which can be completely replaced by
solute atoms [37]. Since it is based on mean field theory, only a single value
of GB segregation concentration can be calculated and it does not take into
account site-specific changes of concentrations within the GB structure. It
neglects interactions between individual atoms, which can significantly
contribute to the segregation behavior [38]. Guggenheim and Fowler extended
the segregation model, taking into account the interaction between neighboring
atoms and the probabilities of neighboring atoms being a solute or matrix
element [39]. Recently, more and more attention was paid to site-specific
segregation within the GB structure [40, 41, 42, 43, 44]. In order to proof
thermodynamic models which take that into account, experimental observations
are needed, which can validate the theoretically calculated GB structure
itself and the segregational site occupancies [23].

Due to its high conductivity, Cu is used in many electronic applications and
long-pulse high field magnets [45]. But because Cu is also quite soft and
would not sustain the electromagnetic forces, alloying different elements is a
common technique to increase its tensile strength which needs to be balanced
with an undesired concomitant decrease in conductivity [46]. When alloying Ag
to Cu, an increase in tensile strength can be observed [7, 47]. Upon specific
heat treatments, it can be accompanied with a high conductivity of
$\geq$80$\%$ IACS (International Annealed Copper Standard) [48, 49, 50].

Even though Cu-Ag alloys are promising materials, only few experimental
studies reveal the segregation of elements at the GB on an atomistic level
[29, 30]. Divinski et al. investigated the diffusion behavior of Ag in Cu
$\Sigma$ 5 GBs and observed abrupt changes in diffusivity at elevated
temperatures, attributed to a GB phase transformation [51, 26]. Simulation
studies on Cu-Ag systems can explain underlying atomistic effects [52, 47, 7,
2, 53, 26]. However, experimental observations are needed to confirm the
atomistic changes and structures at the GB. Cu-Ag is a good model system to
study segregation at GBs due to its extended miscibility gap and absence of
intermetallic phases [54].

Here, we investigate Ag segregation at a $\Sigma$37c $\langle
111\rangle\\{347\\}$ GB at atomic resolution using aberration corrected
scanning transmission electron microscopy (STEM) combined with atomically
resolved energy dispersive X-ray spectroscopy (EDS) and hybrid molecular
dynamics (MD)/Monte Carlo (MC) simulations. We show different methods for
correlating experimental observations with simulations and investigate the
evolution of Ag segregation as a function of its concentration. Finally, we
discuss different parameters which may be responsible for the site occupation
of Ag within a high angle GB.

##  II Methods

###  II.1 Experimental specimen preparation and STEM investigation

A Cu thin film was deposited on a $\langle 0001\rangle$ sapphire wafer by the
Central Scientific Facility Materials of the Max Planck Institute for
Intelligent Systems in Stuttgart. The Al2O3 substrate was sputter cleaned at
$200\text{\,}\mathrm{e}\mathrm{V}$ for
$5\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$ and subsequently annealed at
$None$C for 1h. In a next step, a $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$
thick layer of Cu was deposited at room temperature by molecular beam epitaxy
(MBE) with a rate of $0.3\text{\,}\mathrm{n}\mathrm{m}\mathrm{/}\mathrm{s}$.
After deposition, the film was annealed for $2\text{\,}\mathrm{h}$ at
$450\text{\,}\mathrm{\SIUnitSymbolCelsius}$ inside the same chamber. For the
investigation of Ag segregation, $100\text{\,}\mathrm{n}\mathrm{m}$ of Ag were
sputtered on top of the film at the Ruhr-Universität Bochum. Prior to that,
the top oxide layer was removed by Ar+ sputtering. Consecutively, the film
covered with Ag was annealed at $600\text{\,}\mathrm{\SIUnitSymbolCelsius}$
for $16\text{\,}\mathrm{h}$. For comparison, one part of the same film was
annealed at $600\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for
$16\text{\,}\mathrm{h}$ without a Ag layer. Both films, with and without Ag,
were characterised using a Thermo Fisher Scientific Scios2HiVac dual-beam
secondary electron microscope (SEM) equipped with an electron-backscattered
diffraction (EBSD) detector. Inverse pole figure (IPF) maps of the EBSD scans
helped to identify the GBs of interest, which were then lifted out using a
focused Ga+ ion beam (FIB). The rough milling was performed at
$30\text{\,}\mathrm{k}\mathrm{V}$ and $5\text{\,}\mathrm{n}\mathrm{A}$, going
sequentially down to $5\text{\,}\mathrm{k}\mathrm{V}$ and
$7.7\text{\,}\mathrm{p}\mathrm{A}$ up to a thickness of about
$40\text{\,}\mathrm{n}\mathrm{m}$ to $80\text{\,}\mathrm{n}\mathrm{m}$.

The FIB lamellas were then inspected using a probe-corrected FEI Titan Themis
80-300 (Thermo Fisher Scientific) (scanning) transmission electron microscope
((S)TEM). The TEM is equipped with a high-brightness field emission gun, set
to a voltage of $300\text{\,}\mathrm{k}\mathrm{V}$ and a probe current of
70–$80\text{\,}\mathrm{p}\mathrm{A}$ for imaging. A high-angle annular dark-
field (HAADF) detector (Fishione Instruments Model 3000) was used to record
the HAADF-STEM datasets with collection angles of
78–$200\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$ and a semi-
convergence angle of $17\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$.
EDS measurements were performed at $120\text{\,}\mathrm{k}\mathrm{V}$ and
acquired for at least $15\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$. As we
registered a $120\text{\,}\mathrm{k}\mathrm{V}$ STEM-HAADF image along with
the EDS measurement and wanted to ensure atomic resolution, a fine scan step
size of 512x172 px with a pixel size of $14.47\text{\,}\mathrm{p}\mathrm{m}$
was chosen. For the EDS intensity, subsequent binning of 4x4 pixels was needed
to obtain sufficient data for each position. An additional gaussian filter was
used to make intensity changes in the Ag L-$\alpha$ peak more visible,
ensuring at the same time that the main findings stayed unaffected.

Figure 1: Characterization of the film with and without Ag. a) EBSD of the
microstructure of the pure Cu film. b) SEM image of the surface of the film
after Ag deposition and annealing. c) Bright field-TEM image of a cross
section of a $\Sigma$ 37c GB of the Cu thin film with Ag. d) EDS map from the
interface of the GB with the substrate shows a Ag precipitate at the bottom,
which signifies that Ag diffused entirely across the GB down to the substrate.

###  II.2 MD/MC simulations

We used LAMMPS [55, 56] with an embedded atom method (EAM) potential for Cu–Ag
by Williams et al. [57] for all simulations. In a first step, a bicrystal
containing the experimentally-observed GB was obtained by joining two copper
fcc crystals to produce a $\Sigma 37$c GB with $[\overline{1}11]$ tilt axis
and $(\overline{3}\overline{7}4)$ and $(3\overline{4}7)$ GB planes,
respectively. We varied the relative displacements between the crystallites
followed by minimization of the potential energy until the experimentally-
observed structure was found ($\gamma$-surface method). The final system was
approximately of size $94\times
81\times$187\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{3}$$ with 119 880 atoms.
We applied open boundaries normal to the GB and otherwise periodic boundary
conditions.

In a second step, we computed the bulk silver concentration. Alloying was
simulated in an isobaric semi-grand-canonical ensemble by hybrid MD/MC
simulations [58], defined by the total number of atoms $N$, pressure
$p=$0\text{\,}\mathrm{P}\mathrm{a}$$, temperature $T$, and chemical potential
difference $\Delta\mu=\mu_{\text{Ag}}-\mu_{\text{Cu}}$. The latter controls
the concentration of silver. MD time integration with a time step length of
$2\text{\,}\mathrm{f}\mathrm{s}$ was alternated with $0.1N$ MC trial steps
every 20 MD time steps. The bulk silver concentration was computed by running
the simulations with $\Delta\mu$ from $0.3\text{\,}\mathrm{e}\mathrm{V}$ to
$1.0\text{\,}\mathrm{e}\mathrm{V}$ at $300\text{\,}\mathrm{K}$ and
$500\text{\,}\mathrm{K}$ on a fully periodic system of 32 000 atoms ($20\times
20\times 20$ unit cells) of fcc copper for $10\text{\,}\mathrm{n}\mathrm{s}$.

Finally, we simulated the segregation of silver to the GBs. In order to avoid
size effects due to the excess GB stress [59], we fixed the sizes of the
systems containing the GBs in the periodic directions using the lattice
constant of the pure copper system at the given temperature, thus switching to
the standard isochoric semi-grand-canonical ensemble. This is a reasonable
approximation due to the very low solubility of Ag in Cu, meaning that the
lattice constant is not a function of $\Delta\mu$. The lattice constants were
obtained by equilibrating a periodic, pure fcc Cu system in the
isothermal–isobaric ensemble for $250\text{\,}\mathrm{p}\mathrm{s}$. Then, the
system containing the GB was scaled to the correct lattice constant and
equilibrated in the semi-grand-canonical ensemble for
$20\text{\,}\mathrm{n}\mathrm{s}$ with the same parameters as above. At
$300\text{\,}\mathrm{K}$ we used $\Delta\mu$ values from
$0.30\text{\,}\mathrm{e}\mathrm{V}$ (almost pure copper) to
$0.74\text{\,}\mathrm{e}\mathrm{V}$ (phase transition to silver at
$0.72\text{\,}\mathrm{e}\mathrm{V}$). At $500\text{\,}\mathrm{K}$ we used
$\Delta\mu$ values from $0.40\text{\,}\mathrm{e}\mathrm{V}$ (very low silver
concentration) to $0.68\text{\,}\mathrm{e}\mathrm{V}$ (GB starts to disorder).
Excess properties were calculated as defined in Refs. [60, 59] and simulation
results were visualized with OVITO [61]. The Ag occupation of atomic columns
as viewed from the tilt axis direction was calculated by averaging over a
MD/MC simulation running for an MD time of $1\text{\,}\mathrm{n}\mathrm{s}$.

##  III Results and Discussion

###  III.1 Microstructure of the investigated thin film

We studied the global thin film microstructure and grain boundary evolution by
EBSD and TEM combined with EDS before and after Ag segregation. An inverse
pole figure map of the electron backscattered diffraction dataset of the pure
Cu thin film is shown in Fig. 1a. The out of plane orientation of most grains
is close to $\langle 111\rangle$ as expected for Cu on a single crystalline
$\langle 0001\rangle$ Al2O3 wafer [62, 63]. The Cu grains are oriented in
orientation relationship II, in which the $\\{111\\}\langle 0\bar{1}1\rangle$
direction of the Cu film is oriented parallel to the $(0001)\langle
2\bar{1}\bar{1}0\rangle$ direction of $\alpha$-Al2O3 substrate [63]. The grain
size varies between several $\mathrm{\SIUnitSymbolMicro m}$ up to more than
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$, all of them being larger than the
film thickness ($2\text{\,}\mathrm{\SIUnitSymbolMicro m}$). The thin film was
then divided into several pieces. One piece was annealed at
$600\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for $16\text{\,}\mathrm{h}$ and no
further grain growth could be observed (see Suppl. Fig. Atomic resolution
observations of silver segregation in a [111] tilt grain boundary in copper a
and b). On another piece we deposited a $100\text{\,}\mathrm{n}\mathrm{m}$
layer of Ag. Subsequent annealing under the same condition as for the piece
without Ag deposition let Ag diffuse along the GBs. An SEM image of the film
after Ag deposition and annealing is shown in Fig. 1b. Ag is still present in
most areas of the film (here visible in brighter colors), whereas some grooved
GBs show a Ag-depleted zone around the GB (indicated by arrows). To verify
that Ag diffused along the complete GB, a cross section of a $\Sigma$ 37c GB
was investigated by TEM (Fig. 1c) and EDS (Fig. 1d). A Ag precipitate was
observed at the interface between the Al2O3 substrate and the Cu film at the
GB indicating that Ag diffused along the GB for the entire film thickness.

###  III.2 Observations of the GB structure with and without Ag

Figure 2: The atomic structure of a $\Sigma$ 37c $\langle 111\rangle$
$\\{3\,4\,7\\}$ GB. a) HAADF-STEM image at $300\text{\,}\mathrm{k}\mathrm{V}$
of a GB with a misorientation of $None$ in the pure Cu film. b) MD model of
the zipper structure in pure Cu. The vertical lines indicate the unit cell.
The triangle, circle and square in petrol show CSL points in different
z-heights. c) HAADF-STEM image at $300\text{\,}\mathrm{k}\mathrm{V}$ of a GB
with a misorientation of $None$ from the Ag segregated Cu film.

In a next step, the atomic structure of the grain boundary and the influence
of Ag segregation to it were investigated. Fig. 2a shows an atomically-
resolved HAADF-STEM image of a $\Sigma$37c $\langle 111\rangle$
$\\{3\,4\,7\\}$ GB in pure Cu with a misorientation of $None$. The
misorientation angle is an average of at least 10 different measurements of
angles between different $\langle 220\rangle$ planes of both grains. The
bright spots can be directly correlated to atomic columns. The GB structure
can be described by the structural unit model [64] and divided into two
alternating sub-units: One double-square unit, consisting of 13 atomic
columns, and one trapezoidal unit with 4 atomic columns. The same atomic
arrangement was found in the pure Cu film after annealing (see Suppl. Fig.
Atomic resolution observations of silver segregation in a [111] tilt grain
boundary in copperd). Atomistic simulations of the same GB also exhibit the
same atomic structure (Fig. 2b). The total length of the repeating GB
structure with both sub-units is equivalent to the length of one third of a
$\Sigma$37c $\langle 111\rangle$ $\\{3\,4\,7\\}$ CSL lattice cell. The CSL
lattice of the investigated GB is shown in Suppl. Fig. Atomic resolution
observations of silver segregation in a [111] tilt grain boundary in copper.
After one third, the CSL lattice shows a coincidence site, which is at a
different position along the tilt axis direction than the first coincidence
site, as indicated by different symbols in the Suppl. Fig. Atomic resolution
observations of silver segregation in a [111] tilt grain boundary in copper.
The coincidence lattice sites are as well indicated in petrol in Fig. 2b. In
general, a multiple of the length of the repeating GB structure should always
be equal to the CSL lattice cell (or an integer multiple of it, in our case it
would be 3 times), allowing the repeating GB structure to exist within the
repeating CSL lattice. Otherwise, the strain field and coincidence sites would
always change and a repeating pattern within the GB would be impossible. The
sub-units of the $\Sigma$37c $\langle 111\rangle$ $\\{3\,4\,7\\}$ GB are
similar to the $\Sigma$19b $\langle 111\rangle$ $\\{2\,5\,3\\}$ GB [28], in
which the misorientation angle between both grains is $None$. In that case, a
single square instead of a double-square unit alternates with the same
trapezoidal unit as observed in our case. When increasing the misorientation
angle between both grains up to $None$, we observed a GB structure which
consists only of the square-typed units (see Suppl. Fig. Atomic resolution
observations of silver segregation in a [111] tilt grain boundary in copperc)
similar to a GB with same macroscopic parameters in Aluminum [65]. Thus, the
structural unit model should be valid for this type of symmetric $\langle
111\rangle$ tilt GBs with misorientation angles between $None$ and $None$.

When investigating a $\Sigma$37c $\langle 111\rangle$ $\\{3\,4\,7\\}$ GB in
the Ag annealed Cu film (Fig. 2c), having the same misorientation of $None$,
the same GB structure can be observed as in the case without Ag: The GB
consists of two alternating sub-units, one double-square unit and one
trapezoidal. Neither faceting nor phase transitions can be observed for this
specific high angle GB upon Ag segregation as it was the case for other GBs in
Cu [29, 26]. However, energy dispersive X-ray spectroscopy showed a clear
enrichment of Ag at the GB (see Suppl. Fig. Atomic resolution observations of
silver segregation in a [111] tilt grain boundary in copper). Meiners et al.
investigated Zr segregation at Cu in a similar GB and observed an enrichment
of Zr at asymmetric steps [28]. As visible in Suppl. Fig. Atomic resolution
observations of silver segregation in a [111] tilt grain boundary in copper,
this seems not to be the case for Ag.

Figure 3: a) HAADF-STEM image at $120\text{\,}\mathrm{k}\mathrm{V}$ of a Ag
segregated $\Sigma$37c $\langle 111\rangle$ $\\{3\,4\,7\\}$ GB with a
misorientation of $None$. b) The intensity of the EDS signal of the Ag
L-$\alpha$ peak at $2.98\text{\,}\mathrm{k}\mathrm{e}\mathrm{V}$ has maxima at
each $\sim$$0.9\text{\,}\mathrm{n}\mathrm{m}$ along the GB, coinciding with
the trapezoidal units.

Having a closer look at the Ag segregated GB, the HAADF-STEM contrast of the
atomic columns in the trapezoidal sub-unit seems to be increased (encircled in
purple in Fig. 2c). Thus, we decided to investigate the GB with Ag segregation
by STEM but using a lower acceleration voltage. At
$120\text{\,}\mathrm{k}\mathrm{V}$, the knock-on damage is reduced and less
energy is transferred to the Ag atoms, avoiding a beam-induced change of their
positions during acquisition and thus blurring of the atomic column contrast
[66]. Indeed, a clearer picture can be drawn here (Fig. 3a): One position
within the trapezoidal sub-unit, from now on called ”top position 1”, has a
significantly higher intensity (indicated by purple arrows). As the intensity
in a HAADF-STEM image correlates with the atomic number of an element, this
implies that Ag – which has a higher Z number (47) than Cu (29) – is enriched
on that specific position. In a next step, near-atomic resolution energy
dispersive X-ray spectroscopy was performed. Fig. 3b shows the intensity of
the Ag-L$\alpha$ peak in the EDS dataset along the GB. Therefore, an
integrated Ag line profile was obtained by measuring the Ag-L$\alpha$ peak
intensities along the GB with a width of $1.74\text{\,}\mathrm{n}\mathrm{m}$
using hyperspy [67]. A significant increase in the Ag intensity can be
observed at the trapezoidal unit. This supports the HAADF-STEM findings that
the trapezoidal units are enriched in Ag.

###  III.3 MD/MC simulations and preferential Ag segregation sites

Figure 4: Different thermodynamic properties for increasing chemical potential
differences between Cu and Ag. Whereas a) the free GB energy $\gamma$
decreases, b) the Ag excess $[N_{\text{Ag}}]$ \- or equally the amount of
monolayers covored by Ag atoms - and c) excess volume $[V]$ of the GB increase
with increasing chemical potential difference. Figure 5: Heat maps of the
amount of Ag in each of the atomic columns at 300K are shown for different
chemical potential differences – a) $0.52\text{\,}\mathrm{e}\mathrm{V}$, b)
$0.58\text{\,}\mathrm{e}\mathrm{V}$, and c)
$0.64\text{\,}\mathrm{e}\mathrm{V}$). d) Schematic showing the five positions
within the GB structure to which Ag segregates mostly. e) The fraction of Ag
atoms in each column as labelled in d) increases with a higher chemical
potential difference.

We also performed hybrid MD/MC simulations at $300\text{\,}\mathrm{K}$ with
periodic boundary conditions on the same GB. The chemical potential difference
between Ag and Cu was increased stepwise, acting as a driving force for Ag to
segregate at the GB. In all simulations, as well as in the experiment, we only
observed the zipper structure independent of the silver concentration. In
order to investigate if GB phase transitions occur, we also performed MD/MC
simulations with open boundaries at $900\text{\,}\mathrm{K}$ for
$100\text{\,}\mathrm{n}\mathrm{s}$ with different chemical potential
differences [52]. We could once again only observe the zipper structure in the
$\Sigma 37$c $\langle 111\rangle$ $\\{347\\}$ GB.

We calculated the GB free energy as
$\gamma=\gamma_{0}-\int[N_{\text{Ag}}]\mathrm{d}\Delta\mu$, with $\gamma_{0}$
being the GB free energy without Ag excess [59]. The free energy of the GB
decreases when the chemical potential difference increases (Fig. 4a), which
signifies a stabilization of the GB upon Ag segregation. In contrast to that,
the Ag excess $[N_{\text{Ag}}]$ of the GB and thus the amount of monolayers
occupied by Ag atoms increases (Fig. 4b). To calculate the monolayers, the Ag
excess was divided by the amount of atoms present in one (347) layer
(containing only 3 atoms per CSL unit cell due to the high indices of the
plane) divided by the area of the layer. The excess volume $[V]$ increases as
well as visible in Fig. 4c. A major increase in excess volume is the main
factor for embrittlement during Bi annealing [12, 13, 14]. In the $\Sigma$5
$[001]$ (310) Cu GB, the excess volume tripled upon Bi segregation [68]. In
our case, the GB’s excess volume only doubled, which may help to explain why
Ag segregation does not cause embrittlement [47]. It should be noted that
other factors, like the local changes in atomic bonding of the GB or kinematic
propagation of the crack tip along the GB, also contribute to the
embrittlement of an alloy.

As in the experiment, we can observe that Ag substitutes specific GB sites on
the atomic level. However, in atomistic simulations we are even able to
directly quantify the amount of Ag atoms in each atomic column. In Fig. 5a-c,
the percentage of Ag atoms in each atomic column in the MD/MC simulations is
plotted for 3 different chemical potentials. Even when the Ag concentration in
the GB increases, the GB structure remains unchanged, with Ag just segregating
to new substitutional sites within the structure. At a small chemical
potential difference of $0.52\text{\,}\mathrm{e}\mathrm{V}$ (Fig. 5a), the top
position 1 is the only one which is occupied by Ag atoms with an amount of
$\sim$$22\text{\,}\%$ of Ag out of all atoms in this column. Increasing the
chemical potential difference to $0.58\text{\,}\mathrm{e}\mathrm{V}$ (Fig.
5b), the top position 1 consists already of $88\text{\,}\%$ Ag atoms and the
bottom position 1 of $\sim$$15\text{\,}\%$ Ag atoms. Increasing the difference
further to $0.64\text{\,}\mathrm{e}\mathrm{V}$ (Fig. 5c), the top position 1
is completely occupied by Ag atoms ($99\text{\,}\%$), while the bottom
position 1 is occupied by $\sim$$68\text{\,}\%$ Ag atoms, followed by top
position 3 ($30\text{\,}\%$ Ag atoms), bottom position 2 ($8\text{\,}\%$ Ag
atoms) and top position 2 ($2\text{\,}\%$ Ag atoms). In Fig. 5e, the amount of
Ag at these five specific positions within the GB structure is plotted over
the chemical potential difference. These five positions are the five with
highest Ag concentrations. First, the top position 1 is getting nearly
completely filled, followed by bottom position 1. Then, top position 3, bottom
position 1 and bottom position 2 get filled upon a higher chemical potential
difference. At chemical potentials higher than
$0.68\text{\,}\mathrm{e}\mathrm{V}$, the regular GB structure starts to
collapse and Ag precipitates form. We repeated this analysis with datasets
equilibrated at elevated temperatures ($500\text{\,}\mathrm{K}$ instead of
$300\text{\,}\mathrm{K}$, see Suppl. Fig. Atomic resolution observations of
silver segregation in a [111] tilt grain boundary in copper). They follow the
same trend at slightly lower chemical potential differences.

The experimental determination of local solute concentrations within a GB
structure is to date extremely challenging, but is important for connecting
atomistic characteristics of GBs with their thermodynamic ensemble properties.
Here, we use the obtained data from atomistic modelling under well defined
boundary conditions as a baseline to quantify the atomic scale experimental
observations. Two different approaches were used to find the best match for
our experiment out of the modelled datasets with varying chemical potential
differences. In a first approach, we use the Gibbsian interfacial excess (GIE)
obtained from STEM-EDS as an averaged compositional quantity and in a second
procedure we quantify the atomic column intensity in the HAADF-STEM images,
comparing it to HAADF-STEM image simulations of the modelled datasets with
different chemical potential differences. Both methods are explained in detail
in the Appendix.

In the first approach, the concentration gradient of Ag is determined by STEM-
EDS and averaged across several nanometers (here $\sim$7 nm) of the GB. Then,
the concentration of Ag in both grains is subtracted from the concentration
gradient, since we are only interested in the Ag excess at the GB. In a next
step, the concentration is converted to an atomic density and then integrated
to obtain the cumulative GIE, being $1.67\pm
0.5\text{\,}\mathrm{n}\mathrm{m}^{-2}$. It can then be compared to the ones
from simulations with different amounts of Ag segregation, which were analyzed
in the same way. Simulations with chemical potential differences of 0.55 to
$0.59\text{\,}\mathrm{e}\mathrm{V}$ have a GIE lying in the uncertainty range
of the experimental observations. For lower chemical potential differences,
less Ag segregates to the GB and thus the GIE would be lower. Is the chemical
potential difference higher than $0.59\text{\,}\mathrm{e}\mathrm{V}$, more Ag
is present at the GB and thus the GIE is higher compared to the experimentally
obtained value. It needs to be mentioned that the method is limited due to
channelling effects since the analysis was performed when the sample was in
zone axis, which might change concentration values. Also sample thickness and
background noise can have an influence on concentration calculations.

In the second approach, the atomic column intensity of the Ag columns at the
GB in the HAADF-STEM images is determined and related to the atomistic
simulation by STEM image simulations. After subtracting a polynomial
background from the atomic resolution image, we normalized the intensities of
the Ag columns to the averaged intensity of the Cu atomic columns within a
grain to obtain comparable values. The experimental datasets reveal
intensities of Ag within the top position 1 varying between 1.06 and 1.42
relative to the intensity of Cu atoms within the grain (see Appendix Fig. 8c).
These values are then compared to HAADF-STEM image simulations of the
atomistic models with different chemical potentials. The conditions for the
HAADF-STEM image simulation have to be as close as possible to the real
experimental conditions, including probe size, collection angles and sample
thickness. For chemical potential differences of more than
$0.56\text{\,}\mathrm{e}\mathrm{V}$, the intensity of Ag columns compared to
Cu columns within the grains, is close to the experimentally observed value.
At higher chemical potential differences than
$0.58\text{\,}\mathrm{e}\mathrm{V}$, the intensity of the top position 1 does
not change significantly since at this point, the column is already completely
filled by Ag. However, in higher chemical potential differences, other columns
get occupied with Ag and thus the intensities of these columns would increase.
Apart from one column (see discussion below), this is not the case in our
experiment. Thus, simulations with chemical potential differences higher than
$0.6\text{\,}\mathrm{e}\mathrm{V}$ would not match to our experiment.

To conclude, both methods which can compare simulation and experiment reveal
that simulations with chemical potential differences between
$0.55\text{\,}\mathrm{t}$o $0.59\text{\,}\mathrm{e}\mathrm{V}$ would fit best
to our experimental observations.

In Fig. 3a, one can also observe that different columns of the top position 1
have different HAADF-intensities. These values are normalized intensities
relative to the average intensity of atomic columns within the grains. Knock-
on damage, which can promote the re-distribution of atoms and thus leading to
variations in intensities during longer acquisition times have been minimized
by acquiring the data at $120\text{\,}\mathrm{k}\mathrm{V}$. Other factors,
such as contamination, build-up or sputtering of atoms at the GB can also lead
to errors in the calculation of atomic column intensities. However, the
intensity changes observed varying on a column-by-column basis over a length
of about $7\text{\,}\mathrm{n}\mathrm{m}$ can be as well related to local
changes in Ag concentration. Furthermore, one bottom position 1 is visibly
brighter. It is indicated in Appendix Fig. 8c in the center of the image and
has a value of 1.24. This value is substantially higher than the values of
neighboring bottom positions 1, which have an intensity of 0.96, which
correlates to a brightness intensity of a Ag-free column. This indicates that
only at that specific position, the Ag intensity might be as well increased,
indicating that the bottom position 1 is also partially filled by Ag.
Furthermore, this observation underlines that local intensity changes might be
present.

In the next section, we want to elucidate the reason for Ag occupying these
specific positions.

###  III.4 Influence of GB structure on preferential segregation sites

Figure 6: Properties of different GB sites. a) The segregation energy of each
position. The solid bars represent individual segregation energies and the
hatched bars represent segregation energies when successively filling the
positions (e.g., the hatched green bar shows the segregation energy when top
position 1 and bottom position 1 are already filled). b) Local Voronoi excess
volumes (atomic volume $\Omega$ for each column minus the atomic volume of fcc
atoms $\Omega_{\text{fcc}}$). The absolute volume is shown on the right axis.
For the Ag-free system (gray bars), bottom position 1 has a slightly higher
volume compared to top position 1
($\Delta\Omega=$0.06\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{3}$$). Top
position 1, however, expands to a higher atomic volume
($\Delta\Omega=$-0.15\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{3}$$) upon
insertion of Ag than bottom position 1 (colored bars). c-e) Diagram showing
the radial distances and amount of next atoms for the first three positions
getting occupied. The radial distances are measured in pure Cu GBs. The black
lines indicate the neighbor distances in defect-free fcc. Darker bars show
bond shorter than $3.25\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, which
corresponds roughly to the first neighbor shell. The corresponding
coordination numbers (CN) are also shown. f–h) First neighbor shell of an atom
(yellow) in top position 1 (f), bottom position 1 (g) and top position 3 (h).
The brightness indicates the bond length of the neighbor atom (dark being
$2.55\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ and light being
$3.10\text{\,}\mathrm{\SIUnitSymbolAngstrom}$).

In the simulation, we observed a step-by-step filling of different atomic
sites in the GB when increasing the chemical potential difference. We could
correlate two of the states with our experimental observations, in which one
atomic site (called top position 1) is filled nearly completely and a second
atomic site (bottom position 1) is only partially filled. In the experiment,
we can further observe site-by-site intensity variations in the HAADF
intensity, which might be correlated to local changes in Ag occupancy of each
atomic column. To understand the preferential site occupation of Ag atoms in
the GB structure, we investigated different parameters such as the atomic
volumes, distances to nearest neighbor atoms, Steinhardt parameter [69], and
segregation energies of the different positions.

To this end, we first calculated the segregation energy of each position,
defined as the energy difference of a system with Ag atoms on a GB site
compared to a system with Ag atoms in the bulk (Fig. 6a) [21]. Therefore, we
replaced the complete column of each segregation site in the GB individually
with Ag (solid bars in Fig. 6a) and we could observe a negative segregation
energy for all positions. We also tested segregation energies for implanting
isolated Ag atoms to the investigated positions, but only small segregation
energy differences compared to a completely filled column could be observed
(Supplemental Figs. Atomic resolution observations of silver segregation in a
[111] tilt grain boundary in copper and Atomic resolution observations of
silver segregation in a [111] tilt grain boundary in copper). As expected, the
top position 1, which gets occupied by Ag atoms first in the simulation and is
also occupied by Ag in the experiment, is the one with the lowest segregation
energy, followed by the bottom position 1. Interestingly, it seems that the
top position 2 should be the third column being filled up based on the
segregation energies, but it is the fifth and last column where Ag segregates
to. The hatched bars in Fig. 6(a) show the segregation energies for successive
filling of the columns as it was observed in the simulation, i.e., the
segregation energy of bottom position 1 is calculated for a sample system with
filled top position 1 and so on. For top position 2, the segregation energy
for Ag atoms to this position was calculated in a system in which the other
four columns were already occupied. Under these circumstances, the segregation
energies are consistent with the results in Fig. 5.

As a different potential parameter for predicting preferential segregation
sites, we investigated the free volume of each position in the GB structure
using a Voronoi analysis [70]. Findings for the $\Sigma$5 $\langle 111\rangle$
GB in Cu [12, 71] or $\langle 110\rangle$ GBs in Ni with Pd segregation [72]
suggest that for bigger solute elements, the position in the GB structure with
the highest volume would be the preferential segregation site. With the EAM
potential, we found an excess Voronoi volume of
$0.99\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{3}$ for a substitutional Ag
point defect, indicating that Ag atoms are clearly larger than Cu atoms and
might prefer segregation sites with high free volume. In the investigated GB
structure, the positions with the highest atomic volume in a pure Cu structure
are top position 1 with $13.72\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{3}$ and
bottom position 1 with $13.79\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{3}$,
followed by top position 2 with
$13.21\text{\,}\mathrm{\SIUnitSymbolAngstrom}^{3}$ (Fig. 6b, gray bars). The
bigger atomic volume of top and bottom position 1 compared to the rest of the
sites can explain the preferential segregation to these two sites. Even though
their atomic volumes are similar to each other, the top position 1 is occupied
by Ag atoms prior to bottom position 1.

Differences in atomic strain at different sites can also be a factor for
preferential site occupation [36, 73]. Thus, we investigated the volume
expansion of all five segregation sites upon Ag filling of each site
individually (filled colored bars in Fig. 6b) or upon successive filling
(hatched bars) following the order shown in Fig. 5e. We can observe that top
position 1 has a slightly larger volume than bottom position 1 after being
filled with Ag, even though there was a small tendency in the opposite
direction in the pure Cu case (bottom position 1 was slightly larger compared
to top position 1). This is true regardless whether top position 1 was already
filled by a Ag atom (hatched bar) or only bottom position 1 (colored bar) is
occupied by Ag. It indicates that top position 1 can expand more easily
compared to bottom position 1, which can lead to a preferential position.
However, looking at the atomic volumes of the consecutive positions, the
expected trend of decreasing atomic volumes for less favorable positions
cannot be observed. According to the atomic volumes, top position 2 should be
the position being occupied after top and bottom position 1. However, it is
top position 3, with the lowest atomic volume both in a pure Cu cell and when
it is occupied by Ag. Thus, the atomic volume alone is insufficient to explain
the preferential segregation sites.

In a next step, we also had a look at the specific atomic environments of
different positions. Therefore, we investigated the bond length of all
neighboring atoms as shown in Fig. 6c–h for the first three positions. In
plots c–e it is shown that the closest atoms are present in bond distances
between $2.4\text{/}2.5\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ to
$3.1\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. In none of the three
investigated positions, atoms occur with bond distances of 3.1 to
$3.4\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, but only again at higher
distances than $3.4\text{\,}\mathrm{\SIUnitSymbolAngstrom}$. In a bulk Cu
lattice, the nearest neighbor atoms have a distance of
$2.55\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ (indicated by a black line). In
bottom position 1, two atoms are even slightly closer than that. This might be
unfavorable for Ag due to its atomic size mismatch, but due to the delocalized
nature of the metallic bond it is difficult to draw definite conclusions from
the first neighbor shell environment alone. All in all, no striking,
qualitative differences between the three diagrams are visible that could
serve as a simple predictor for preferential segregation sites.

Furthermore, Scheiber et al. found a linear correlation between the 5th
Steinhardt order parameter [69] and the segregation energy [74]. We therefore
also tried this analysis (Supplemental Fig. Atomic resolution observations of
silver segregation in a [111] tilt grain boundary in copper, calculated using
the software pyscal [75]), but could find no such correlation for any of the
first 12 parameters. This suggests that the changes in the bonding environment
are relatively delocalized in our case so that they are not easily detectable
with methods that consider only the next-neighbor environment. Therefore, the
bond lengths and number of nearest neighbor atoms or the shape of the
polyhedron are not suitable as simple predictors to determine preferred
segregation sites within a GB in a metallic system in accordance with the
observations of Ziebarth et al. for Fe impurities in Si GBs [76].

Furthermore, preferential sites of solutes were also identified due to
dangling bonds or changes of covalent bonds especially in ceramic materials or
semimetals [77, 78, 79, 80]. Upon changes in the electronic structure, the
cohesive energy also changes and can thus lead to embrittlement. Pronounced
effects can be observed if the solute atoms differ in electronegativity
compared to the host element [81, 82]. However, for Ag in Cu, both elements
have similar electronegativity and embrittlement of Cu due to Ag segregation
has not been observed. Since both are metals, no dangling or covalent bonds
are expected and changes in the electronic band structure are not accessible
with the methods used in the present work.

We can conclude that besides the segregation energy, no simple predictor of
the preferential segregation sites could be found. While local free volume
plays a role, local bonding effects cannot be neglected and depend on the
complex atomic environment of the GB. However, these are not easy to compute
and thus not suitable for simple predictions.

##  IV Conclusion

We investigated the segregation behavior of Ag in a symmetric tilt GB in Cu
and different parameters which might affect the site occupation combining
experimental observations and atomistic simulations. The atomic structure of a
$\Sigma$37c $\langle 111\rangle$ $\\{3\,4\,7\\}$ GB in copper was resolved
using HAADF-STEM. The same structure was replicated in MD simulations. After
Ag alloying, Ag atoms only substitute specific positions in the GB. Neither
faceting nor phase transition was observed. Comparing experiment and hybrid
MD/MC simulations using HAADF-STEM intensity changes and Gibbsian interfacial
excess determination, we conclude that simulations with a Ag excess of
$0.015\text{\,}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}\mathrm{\SIUnitSymbolAngstrom}^{-2}$
to
$0.019\text{\,}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}\mathrm{\SIUnitSymbolAngstrom}^{-2}$
match best to our experiment (equivalent to a chemical potential difference of
$\Delta\mu=$ $0.56\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ to
$0.58\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$). In these cases, two columns in
the GB structure are occupied with Ag, whereas top position 1 is mostly filled
and bottom position 1 only partially. In the experiment, local changes of Ag
occupation between columns are observed, which can induce site-specific
changes in GB properties. We investigated the relevance of different
parameters (volume, configuration of surrounding atoms, Steinhardt order
parameter) for predicting preferential segregation sites in the GB structure.
The preferential segregation sites seem not be solely determined by their
local volume of each site and their expansion upon segregation. Also the bond
lengths of surrounding atoms and the Steinhardt order parameter do not show a
simple clear trend following the order in which the positions are filled up
according to simulations. Only the calculation of the segregation energy,
which takes into account the local atomic bonding in each position, follows
the same trend as the occupation of Ag in specific sites. A combination of
both experiment and simulation as presented above is needed to determine the
atomic level distribution of solutes at GBs and for understanding the
atomistic origin of GB segregation. These two ingredients ultimately provide
the link to the thermodynamic characteristics of the interface and how they
affect macroscopic material properties.

##  V Data availability

The main datasets of this study are published under
https://doi.org/10.5281/zenodo.7372733.

##  VI Acknowledgements

We gratefully acknowledge G. Richter’s team from the Max Planck Institute for
Intelligent Systems for producing the Cu thin film and T. Oellers for the Ag
deposition at the Ruhr-Universität Bochum. The authors thank N. Peter for
helpful discussions and initiating this project by supporting in Ag deposition
on the film. 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).

Author contributions: T.B. and L.L. contributed equally to this work. L.L.
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. The simulations and analyses of the simulation data were
performed by T.B.. G.R. provided the Cu thin film, the main material used in
this study. The project was supervised by C.H.L. and G.D., who also
contributed to discussions. G.D. secured funding for L.L. and T.B. via the ERC
grant GB-CORRELATE. L.L. and T.B. prepared the initial draft and all authors
contributed to the preparation of the final manuscript.

##  Appendix A Comparison of experiment and simulation to quantify the Ag
excess

We used two methods to determine the local Ag excess at the GB by comparing
experiment with atomistic simulations. In the first method, we determine the
Gibbsian Interfacial Excess (GIE) from the STEM-EDS and compared it to the GIE
obtained from MD/MC simulations. In the second method, we measure the atomic
column intensity in HAADF-STEM images of the top position 1 in the GB relative
to the bulk Cu columns and compared it to HAADF-STEM image simulations using
the MD/MC cells as input.

### Method 1: Determining the Gibbsian interfacial excess from STEM-EDS

Figure 7: Comparing the calculated gibbsian interfacial excess (GI) value of
simulations with the experiment. a) HAADF-STEM image with EDS signal of Ag
overlaid. The concentration profile of Ag was measured along the arrow, with a
width equal to the frame size. b) Diagrams showing the steps needed to convert
the concentration to the GIE. c) Comparison of the GIE of simulations with
experiments. All simulation datasets for the different chemical potentials
were equilibrated and performed at $300\text{\,}\mathrm{K}$.

We determine the Gibbsian interfacial excess (GIE) from STEM-EDS datasets
similar to the procedure described in Refs. [83, 84]. We used the EDS maps
acquired at $120\text{\,}\mathrm{k}\mathrm{V}$ to reduce knock-on damage and
to maximize the X-ray yield. In a first step, the Ag content was quantified
using the Velox Software (vers. 3.5). For the quantification, a standard-less
Cliff-Lorimer quantification with absorption correction was used, which takes
into account the actual geometry of the detectors and the stage tilt. The
sample thickness was set to $50\text{\,}\mathrm{n}\mathrm{m}$ and a Brown-
Powell ionization cross-section model was used. In a second step, the Ag
concentration profile across a GB with a width of
$7\text{\,}\mathrm{n}\mathrm{m}$ parallel to the GB plane – equivalent to
about 7 structural units of the $\Sigma$ 37c $\langle 111\rangle\\{347\\}$ GB
– was summed. Such a concentration profile is shown in Fig. 7b on the left
side. Since we are only interested in the Ag excess at the GB, thus the
surplus of Ag content, we subtracted the Ag concentration in the grains. In
order to access the GIE, the Ag concentration needs to be converted to an
atomic density. Therefore, the density of fcc Cu along the $\langle
111\rangle$ direction was used, being
$84.67\text{\,}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}\mathrm{n}\mathrm{m}^{-2}$
for a lattice parameter of $a=$3.615\text{\,}\mathrm{\SIUnitSymbolAngstrom}$$.
This is shown in Fig. 7b in the middle. The integral of the atomic density is
then equal to the GIE (Fig. 7b right side). An experimental value of the GIE
of about $1.67\pm 0.5\text{\,}\mathrm{n}\mathrm{m}^{-2}$ was experimentally
determined for 2 independent EDS data sets acquired at different GB positions.

For estimating the errors present in our calculation, the averaged Cu density
as well as the integration width were changed. Thereby, changing the atomic
density by 6$\%$ can lead to an uncertainty in the GIE of up to 22$\%$. A
decrease of 30$\%$ of the integration width can lead to a decrease in GIE of
15 $\%$. Furthermore, the quantification of Ag concentration by EDS has an
uncertainty of $\pm 2\,\text{at-\%}$. To conclude, the method presented above
is sensitive to many parameters affecting its accuracy to determine the GIE.
Thus, we used the experimentally measured GIE only as a rough guideline to
evaluate which range of simulated Ag excess values and hence chemical
potential difference match to the experiment. Therefore, we performed the same
analysis as described above for the simulated datasets, even though the GIE
could be extracted from the simulated datasets in a direct manner as shown in
Fig. 4b. We compared the directly obtained GIE for Ag in the simulated
datasets with the ones calculated analogous to the experiment and observed
deviations of max. 5$\%$. This shows that the pathway of obtaining the GIE is
fairly correct and only experimental uncertainties (like the absolute
quantification of Ag and defining the integration width of the Ag peak) can
affect the experimentally extracted GIE. As expected, with increase of a
chemical potential difference, the GIE increases since more Ag diffuses into
the system and also more Ag is present at the GB. The simulations overlap with
the experiments for chemical potential differences between
$0.55\text{\,}\mathrm{e}\mathrm{V}$ to $0.59\text{\,}\mathrm{e}\mathrm{V}$.

### Method 2: Quantifying the intensity of Ag columns from atomic resolution
using the HAADF-STEM images

In order to estimate the amount of Ag in atomic columns within the GB, we
quantify the atomic column contrast in the HAADF-STEM images. Due to the
complex atomic arrangements of atoms at the GB core, we determine the
intensity of the Ag-rich columns relative to atomic columns within the grain
as a reference. The experimental datasets are then compared to HAADF-STEM
image simulations using the MC/MD simulation cells as input. In a first step,
a 2D-polynomial background subtraction was performed on the experimental
images (Fig. 8a). Then, a peak finding algorithm of the python package ATOMAP
(vers. 0.3.1) was used to determine the position of each of the atomic columns
and their single peak intensity values [85]. The intensity values got
normalized by the average of the intensity of atomic columns located in both
grains (see Fig. 8b). In a final step, the intensity of the top position 1
within the GB structure was extracted. The intensities in the experimental
image vary between 1.06 and 1.39, excluding the minimum and maximum since they
might stem from non-accurate background substraction. Thus, in average, the
top position 1 has an intensity of $1.18\pm 0.15$ as shown in Fig. 8c. Besides
of one bottom position 1, which has an intensity value of 1.24, the intensity
of all other positions does not show a substantial increase from the averaged
value of 1 and are thus not listed here. This method also exhibits several
uncertainties, as the intensity value might be influenced in the first place
by the sample thickness, amorphous layers on the surface of the sample or
oxides and background noise, and in the second step by using different
background correction methods or extracting the intensity of the atomic
columns with other algorithms. Thus, great care has to be taken and the final
results can be only understood as a rough estimation.

Figure 8: Comparing the HAADF-STEM intensities of atomic columns enriched with
Ag with the HAADF-STEM intensity of Cu columns in the grain. a) Intensity of a
background corrected HAADF-STEM image. b) The intensity values of all columns
were measured and normalized by the average intensity values of atomic columns
in the grains. c) Intensity values of the Ag enriched columns. d) A comparison
was performed for the experimental image with HAADF-STEM image simulations,
using the same imaging conditions as used in the experiment. A thickness range
for the simulation cells was used to take into account uncertainties for the
experimentally determined thickness value of the sample.

For the comparison with MD/MC simulations, a simulation cell of a similar
thickness as the lamella in the experiment is needed. We estimated the
thickness of the investigated sample to be between 55 and
$64\text{\,}\mathrm{n}\mathrm{m}$ using EELS and the thickness determination
according to Malis et al. [86]. In a next step, we took the MC/MD cells and
replicated them periodically along the $\langle 111\rangle$ direction to the
desired thickness. In a next step, we used these cells to create synthetic
HAADF-STEM images using the multislice algorithm with parameters which match
the experimental parameters as good as possible (an aberration-free electron
probe with $120\text{\,}\mathrm{k}\mathrm{V}$, semi-angle of
$24\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$, focal spread of
$60\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, an annular detector detecting
with a collection angle range of 95 to
$200\text{\,}\mathrm{m}\mathrm{r}\mathrm{a}\mathrm{d}$, corresponding to the
setting of our HAADF detector in the experiment, and a beam scan step size of
$0.14\text{\,}\mathrm{\SIUnitSymbolAngstrom}$, which corresponds to the pixel
size). The image simulation was performed with the python library abTEM vers.
1.0.0b11 [87].

Having the simulated HAADF-STEM images for each chemical potential, we
followed the same steps as for the experimental images. Only the first step of
background correction was not necessary since the contribution from background
is negligible in simulation. All atomic column positions and their peak
intensities were determined using ATOMAP. The intensity of all columns was
normalized by the average intensity of the atomic columns within the bulk
grains. In Fig. 8d, the change in intensity ratio of the top position 1 in
dependence of chemical potential difference is shown for different simulation
cell thicknesses. The average intensity ratio determined from the experiment
is given as black line and the gray shaded area represents the uncertainty
range considering different columns. The intensity values of top position 1
from the simulated images start to overlap with the ones from experiments for
chemical potential differences at about $0.56\text{\,}\mathrm{e}\mathrm{V}$ to
$0.58\text{\,}\mathrm{e}\mathrm{V}$, depending on the simulated thickness.
When increasing the chemical potential difference further, we can see that
other positions (as bottom position 1, 2 and top position 2,3), get occupied
by Ag and thus increase in intensity values. In the experiment, the HAADF STEM
intensities of top position 2,3 and bottom position 1,2 do not show
significantly higher values than other positions within the GB (besides bottom
posiiton 1 in one position as indicated in Fig. 8c). For simulated datasets
with a chemical potential difference of more than
$0.60\text{\,}\mathrm{e}\mathrm{V}$, a significant increase in HAADF STEM
intensity in other atomic columns is detected. Thus, when comparing HAADF-STEM
intensities of simulations and experiment, simulations using a chemical
potential difference of $0.56\text{\,}\mathrm{e}\mathrm{V}$ and
$0.60\text{\,}\mathrm{e}\mathrm{V}$ are matching best with the experiments.

To conclude, both independent methods predict that a chemical potential
difference between $0.56\text{\,}\mathrm{e}\mathrm{V}$ and
$0.60\text{\,}\mathrm{e}\mathrm{V}$ would fit best to our experimental
dataset.

## References

  * Özerinç  _et al._ [2012] S. Özerinç, K. Tai, N. Q. Vo, P. Bellon, R. S. Averback, and W. P. King, Grain boundary doping strengthens nanocrystalline copper alloys, Scripta Materialia 67, 720 (2012).
  * Koju and Mishin [2020a] R. K. Koju and Y. Mishin, Atomistic study of grain-boundary segregation and grain-boundary diffusion in al-mg alloys, Acta Materialia 201, 596 (2020a).
  * Basu  _et al._ [2016] I. Basu, K. G. Pradeep, C. Mießen, L. A. Barrales-Mora, and T. Al-Samman, The role of atomic scale segregation in designing highly ductile magnesium alloys, Acta Materialia 116, 77 (2016).
  * Khang  _et al._ [2006] D. Y. Khang, H. Jiang, Y. Huang, and J. A. Rogers, A stretchable form of single-crystal silicon for high-performance electronics on rubber subtrates, Science 311, 208 (2006).
  * Schäfer and Albe [2012] J. Schäfer and K. Albe, Influence of solutes on the competition between mesoscopic grain boundary sliding and coupled grain boundary motion, Scripta Materialia 66, 315 (2012).
  * Rupert  _et al._ [2011] T. J. Rupert, J. C. Trenkle, and C. A. Schuh, Enhanced solid solution effects on the strength of nanocrystalline alloys, Acta Materialia 59, 1619 (2011).
  * Vo  _et al._ [2011] N. Q. Vo, J. Schäfer, R. S. Averback, K. Albe, Y. Ashkenazy, and P. Bellon, Reaching theoretical strengths in nanocrystalline Cu by grain boundary doping, Scripta Materialia 65, 660 (2011).
  * Wei and Grabke [1986] W. Wei and H. J. Grabke, The effect of alloying elements on the grain boundary segregation of phosphorus in iron and the intergranular corrosion of the Fe-P system, Corrosion Science 26, 223 (1986).
  * Seah [1980] M. P. Seah, Adsorption-induced interface decohesion, Acta Metallurgica 28, 955 (1980).
  * Lejcek [2010] P. Lejcek,  _Grain boundary segregation in metals_ , Vol. 136 (Springer Science & Business Media, 2010).
  * Hampe [1874] W. Hampe, Beiträge zu der metallurgie des kupfers, Zeitschrift für das Berg-, Hütten-und Salinenwesen im Deutschen Reich 22, 93 (1874).
  * Duscher  _et al._ [2004] G. Duscher, M. F. Chisholm, U. Alber, and M. Rühle, Bismuth-induced embrittlement of copper grain boundaries, Nature Materials 2004 3:9 3, 621 (2004).
  * Keast  _et al._ [1998] V. J. Keast, J. Bruley, P. Rez, J. M. Maclaren, and D. B. Williams, Chemistry and bonding changes associated with the segregation of bi to grain boundaries in cu, Acta Materialia 46, 481 (1998).
  * Schweinfest  _et al._ [2004] R. Schweinfest, A. T. Paxton, and M. W. Finnis, Bismuth embrittlement of copper is an atomic size effect, Nature 2004 432:7020 432, 1008 (2004).
  * Ainslie  _et al._ [1960] N. G. Ainslie, R. E. Hoffman, and A. U. Seybolt, Sulfur segregation at $\alpha$-iron grain boundaries–I, Acta Metallurgica 8, 523 (1960).
  * Thomas and Chalmers [1955] W. R. Thomas and B. Chalmers, The segregation of impurities to grain boundaries, Acta Metallurgica 3, 17 (1955).
  * Li and Zhang [1995] G. H. Li and L. D. Zhang, Relationship between misorientation and bismuth induced embrittlement of [001] tilt boundary in copper bicrystal, Scripta Metallurgica et Materialia 32, 1335 (1995).
  * Powell and Mykura [1973] B. D. Powell and H. Mykura, The segregation of bismuth to grain boundaries in copper-bismuth alloys, Acta Metallurgica 21, 1151 (1973).
  * Hondros [1975] E. D. Hondros, Grain boundary segregation. the current situation and future requirements, Le Journal de Physique Colloques 36, 117 (1975).
  * Watanabe [1985] T. Watanabe, Structural effects on grain boundary segregation, hardening and fracture, Le Journal de Physique Colloques 46, 555 (1985).
  * Vitek and Wang [1982] V. Vitek and G. J. Wang, Atomic structure of grain boundaries and intergranular segregation, Journal de Physique (Paris), Colloque 43, 147 (1982).
  * Wynblatt and Chatain [2006] P. Wynblatt and D. Chatain, Anisotropy of segregation at grain boundaries and surfaces, Metallurgical and Materials Transactions A 2006 37:9 37, 2595 (2006).
  * Lejček  _et al._ [2018] P. Lejček, M. Všianská, and M. Šob, Recent trends and open questions in grain boundary segregation, Journal of Materials Research 33, 2647 (2018).
  * Raabe  _et al._ [2014] D. Raabe, M. Herbig, S. Sandlöbes, Y. Li, D. Tytko, M. Kuzmina, D. Ponge, and P. P. Choi, Grain boundary segregation engineering in metallic alloys: A pathway to the design of interfaces, Current Opinion in Solid State and Materials Science 18, 253 (2014).
  * Cantwell  _et al._ [2020] P. R. Cantwell, T. Frolov, T. J. Rupert, A. R. Krause, C. J. Marvel, G. S. Rohrer, J. M. Rickman, and M. P. Harmer, Grain boundary complexion transitions, Annual Review of Materials Research 50, 465 (2020).
  * Frolov  _et al._ [2015] T. Frolov, M. Asta, and Y. Mishin, Segregation-induced phase transformations in grain boundaries, Physical Review B - Condensed Matter and Materials Physics 92, 10.1103/PhysRevB.92.020103 (2015).
  * Ference and Balluffi [1988] T. G. Ference and R. W. Balluffi, Observation of a reversible grain boundary faceting transition induced by changes of composition, Scripta Metallurgica 22, 1929 (1988).
  * Meiners  _et al._ [2020] T. Meiners, J. M. Duarte, G. Richter, G. Dehm, and C. H. Liebscher, Tantalum and zirconium induced structural transitions at complex [111] tilt grain boundaries in copper, Acta Materialia 190, 93 (2020).
  * Peter  _et al._ [2018] N. J. Peter, T. Frolov, M. J. Duarte, R. Hadian, C. Ophus, C. Kirchlechner, C. H. Liebscher, and G. Dehm, Segregation-induced nanofaceting transition at an asymmetric tilt grain boundary in copper, Physical Review Letters 121, 10.1103/PhysRevLett.121.255502 (2018).
  * Peter  _et al._ [2021] N. J. Peter, M. J. Duarte, C. Kirchlechner, C. H. Liebscher, and G. Dehm, Faceting diagram for ag segregation induced nanofaceting at an asymmetric cu tilt grain boundary, Acta Materialia 214, 116960 (2021).
  * Feng  _et al._ [2016] B. Feng, T. Yokoi, A. Kumamoto, M. Yoshiya, Y. Ikuhara, and N. Shibata, Atomically ordered solute segregation behaviour in an oxide grain boundary, Nature Communications 2016 7:1 7, 1 (2016).
  * Ma  _et al._ [2013] S. Ma, P. R. Cantwell, T. J. Pennycook, N. Zhou, M. P. Oxley, D. N. Leonard, S. J. Pennycook, J. Luo, and M. P. Harmer, Grain boundary complexion transitions in WO3\- and CuO-doped TiO2 bicrystals, Acta Materialia 61, 1691 (2013).
  * Yoon  _et al._ [2017] H. I. Yoon, D. K. Lee, H. B. Bae, G. Y. Jo, H. S. Chung, J. G. Kim, S. J. L. Kang, and S. Y. Chung, Probing dopant segregation in distinct cation sites at perovskite oxide polycrystal interfaces, Nature Communications 2017 8:1 8, 1 (2017).
  * Li  _et al._ [2014] C. X. Li, S. H. Dang, L. P. Wang, C. L. Zhang, and P. D. Han, First principles investigation into effects of Cr on segregation of S or Cl at $\alpha$-Fe $\Sigma$5(210) grain boundary (Maney Publishing, 2014) pp. S41012–S41016.
  * Parajuli  _et al._ [2019] P. Parajuli, D. Romeu, V. Hounkpati, R. Mendoza-Cruz, J. Chen, M. J. Yacamán, J. Flowers, and A. Ponce, Misorientation dependence grain boundary complexions in symmetric tilt al grain boundaries, Acta Materialia 181, 216 (2019).
  * Nie  _et al._ [2013] J. F. Nie, Y. M. Zhu, J. Z. Liu, and X. Y. Fang, Periodic segregation of solute atoms in fully coherent twin boundaries, Science 340, 957 (2013).
  * McLean [1957] D. McLean,  _Grain boundaries in metals_ (Clarendon Press Oxford, 1957) pp. 1–346.
  * Hondros and Seah [1977] E. D. Hondros and M. P. Seah, The theory of grain boundary segregation in terms of surface adsorption analogues, Metallurgical Transactions A 1977 8:9 8, 1363 (1977).
  * Fowler [1939] R. H. Fowler,  _Statistical thermodynamics_ (CUP Archive, 1939).
  * White and Coghlan [1977] C. L. White and W. A. Coghlan, The spectrum of binding energies approach to grain boundary segregation, Metallurgical Transactions A 1977 8:9 8, 1403 (1977).
  * White and Stein [1978] C. L. White and D. F. Stein, Sulfur segregation to grain boundaries in Ni3Al and Ni3(Al,Ti) alloys, Metallurgical and Materials Transactions A 1978 9:1 9, 13 (1978).
  * Udler and Seidman [1998] D. Udler and D. N. Seidman, Solute segregation at [001] tilt boundaries in dilute f.c.c. alloys, Acta Materialia 46, 1221 (1998).
  * Wagih and Schuh [2020] M. Wagih and C. A. Schuh, Grain boundary segregation beyond the dilute limit: Separating the two contributions of site spectrality and solute interactions, Acta Materialia 199, 63 (2020).
  * Huber  _et al._ [2018] L. Huber, R. Hadian, B. Grabowski, and J. Neugebauer, A machine learning approach to model solute grain boundary segregation, npj Computational Materials 2018 4:1 4, 1 (2018).
  * Debray and Frings [2013] F. Debray and P. Frings, State of the art and developments of high field magnets at the “laboratoire national des champs magnétiques intenses”, Comptes Rendus Physique 14, 2 (2013).
  * Zherlitsyn  _et al._ [2012] S. Zherlitsyn, B. Wustmann, T. Herrmannsdorfer, and J. Wosnitza, Status of the pulsed-magnet-development program at the dresden high magnetic field laboratory, IEEE Transactions on Applied Superconductivity 22, 10.1109/TASC.2012.2182975 (2012).
  * Lozovoi  _et al._ [2006] A. Y. Lozovoi, A. T. Paxton, and M. W. Finnis, Structural and chemical embrittlement of grain boundaries by impurities: A general theory and first-principles calculations for copper, Physical Review B 74, 155416 (2006).
  * Sakai  _et al._ [1991] Y. Sakai, K. Inoue, T. Asano, H. Wada, and H. Maeda, Development of high‐strength, high‐conductivity cu–ag alloys for high‐field pulsed magnet use, Applied Physics Letters 59, 2965 (1991).
  * Sakai and Schneider-Muntau [1997] Y. Sakai and H. J. Schneider-Muntau, Ultra-high strength, high conductivity cu-ag alloy wires, Acta Materialia 45, 1017 (1997).
  * Yang  _et al._ [2020] H. Y. Yang, Z. C. Ma, C. H. Lei, L. Meng, Y. T. Fang, J. B. Liu, and H. T. Wang, High strength and high conductivity cu alloys: A review, Science China Technological Sciences 2020 63:12 63, 2505 (2020).
  * Divinski  _et al._ [2012] S. V. Divinski, H. Edelhoff, and S. Prokofjev, Diffusion and segregation of silver in copper $\Sigma$5(310) grain boundary, Physical Review B - Condensed Matter and Materials Physics 85, 144104 (2012).
  * Frolov  _et al._ [2013] T. Frolov, D. L. Olmsted, M. Asta, and Y. Mishin, Structural phase transformations in metallic grain boundaries, Nature Communications 4, 1899 (2013).
  * Koju and Mishin [2020b] R. K. Koju and Y. Mishin, Relationship between grain boundary segregation and grain boundary diffusion in cu-ag alloys, Physical Review Materials 4, 073403 (2020b).
  * Subramanian and Perepezko [1993] P. R. Subramanian and J. H. Perepezko, The ag-cu (silver-copper) system, Journal of Phase Equilibria 14, 62 (1993).
  * Plimpton [1995] S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, Journal of Computational Physics 117, 1 (1995).
  * Thompson  _et al._ [2022] A. P. Thompson, H. M. Aktulga, R. Berger, D. S. Bolintineanu, W. M. Brown, P. S. Crozier, P. J. in ’t Veld, A. Kohlmeyer, S. G. Moore, T. D. Nguyen, R. Shan, M. J. Stevens, J. Tranchida, C. Trott, and S. J. Plimpton, LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales, Comput. Phys. Commun. 271, 108171 (2022), https://lammps.org/.
  * Williams  _et al._ [2006] P. L. Williams, Y. Mishin, and J. C. Hamilton, An embedded-atom potential for the Cu–Ag system, Modell. Simul. Mater. Sci. Eng. 14, 817 (2006).
  * Sadigh  _et al._ [2012] B. Sadigh, P. Erhart, A. Stukowski, A. Caro, E. Martinez, and L. Zepeda-Ruiz, Scalable parallel Monte Carlo algorithm for atomistic simulations of precipitation in alloys, Phys. Rev. B 85, 184203 (2012).
  * Frolov and Mishin [2012a] T. Frolov and Y. Mishin, Thermodynamics of coherent interfaces under mechanical stresses. II. Application to atomistic simulation of grain boundaries, Phys. Rev. B 85, 224107 (2012a).
  * Frolov and Mishin [2012b] T. Frolov and Y. Mishin, Thermodynamics of coherent interfaces under mechanical stresses. I. Theory, Phys. Rev. B 85, 224106 (2012b).
  * Stukowski [2010] A. Stukowski, Visualization and analysis of atomistic simulation data with OVITO – the Open Visualization Tool, Modell. Simul. Mater. Sci. Eng. 18, 015012 (2010), https://ovito.org/.
  * Katz [1968] G. Katz, The epitaxy of copper on sapphire, Applied Physics Letters 12, 161 (1968).
  * Dehm  _et al._ [2005] G. Dehm, H. Edongué, T. Wagner, S. H. Oh, and E. Arzt, Obtaining different orientation relationships for cu films grown on (0001) $\alpha$-al2o3 substrates by magnetron sputtering, Zeitschrift für Metallkunde 96, 249 (2005), doi: 10.3139/146.101027.
  * Han  _et al._ [2017] J. Han, V. Vitek, and D. J. Srolovitz, The grain-boundary structural unit model redux, Acta Materialia 133, 186 (2017).
  * Ahmad  _et al._ [2023] S. Ahmad, T. Brink, C. H. Liebscher, and G. Dehm, Microstates and defects of incoherent $\Sigma$3 [111] twin boundaries in aluminum, Acta Materialia 243, 118499 (2023).
  * Egerton  _et al._ [2004] R. F. Egerton, P. Li, and M. Malac, Radiation damage in the tem and sem, Micron 35, 399 (2004).
  * de la Peña  _et al._ [2021] F. de la Peña, E. Prestat, V. T. Fauske, P. Burdet, J. Lähnemann, T. Furnival, P. Jokubauskas, M. Nord, T. Ostasevicius, K. E. MacArthur, D. N. Johnstone, M. Sarahan, T. Aarholt, J. Taillon, pquinn dls, V. Migunov, A. Eljarrat, J. Caron, T. Poon, S. Mazzucco, C. Francis, B. Martineau, actions user, S. Somnath, T. Slater, N. Tappy, M. Walls, N. Cautaerts, F. Winkler, and DENSmerijn, hyperspy/hyperspy: Release v1.6.5 10.5281/ZENODO.5608741 (2021).
  * Lozovoi and Paxton [2008] A. Y. Lozovoi and A. T. Paxton, Boron in copper: A perfect misfit in the bulk and cohesion enhancer at a grain boundary, Physical Review B - Condensed Matter and Materials Physics 77, 165413 (2008).
  * Steinhardt  _et al._ [1983] P. J. Steinhardt, D. R. Nelson, and M. Ronchetti, Bond-orientational order in liquids and glasses, Phys. Rev. B 28, 784 (1983).
  * Voronoi [1908] G. Voronoi, Nouvelles applications des paramètres continus à la théorie des formes quadratiques. deuxième mémoire. recherches sur les parallélloèdres primitifs., Journal für die reine und angewandte Mathematik (Crelles Journal) 1908, 198 (1908).
  * Huang  _et al._ [2020] Z. Huang, P. Wang, F. Chen, Q. Shen, and L. Zhang, Understanding solute effect on grain boundary strength based on atomic size and electronic interaction, Scientific Reports 2020 10:1 10, 1 (2020).
  * Rittner and Seidman [1997] J. D. Rittner and D. N. Seidman, Solute-atom segregation to $\langle$110$\rangle$ symmetric tilt grain boundaries, Acta Materialia 45, 3191 (1997).
  * Käshammer and Sinno [2015] P. Käshammer and T. Sinno, A mechanistic study of impurity segregation at silicon grain boundaries, Journal of Applied Physics 118, 10.1063/1.4929637 (2015).
  * Scheiber  _et al._ [2015] D. Scheiber, V. I. Razumovskiy, P. Puschnig, R. Pippan, and L. Romaner, Ab initio description of segregation and cohesion of grain boundaries in W–25 at.$\%$ Re alloys, Acta Materialia 88, 180 (2015).
  * Menon  _et al._ [2019] S. Menon, G. Díaz Leines, and J. Rogal, pyscal: A python module for structural analysis of atomic environments, J. Open Source Softw. 4, 1824 (2019).
  * Ziebarth  _et al._ [2015] B. Ziebarth, M. Mrovec, C. Elsässer, and P. Gumbsch, Interstitial iron impurities at grain boundaries in silicon: A first-principles study, Physical Review B - Condensed Matter and Materials Physics 91, 035309 (2015).
  * Zhao and Li [2019] D. Zhao and Y. Li, Revealing the factors influencing grain boundary segregation of p, as in si: Insights from first-principles, Acta Materialia 168, 52 (2019).
  * Buban  _et al._ [2006] J. P. Buban, K. Matsunaga, J. Chen, N. Shibata, W. Y. Ching, T. Yamamoto, and Y. Ikuhara, Grain boundary strengthening in alumina by rare earth impurities, Science 311, 212 (2006).
  * Fabris and Elsässer [2003] S. Fabris and C. Elsässer, First-principles analysis of cation segregation at grain boundaries in $\alpha$-Al2O3, Acta Materialia 51, 71 (2003).
  * Chen  _et al._ [2005] J. Chen, Y. N. Xu, P. Rulis, L. Ouyang, and W. Y. Ching, Ab initio theoretical tensile test on Y-doped ${\Sigma}=3$ grain boundary in $\alpha$-Al2O3, Acta Materialia 53, 403 (2005).
  * Briant [1990] C. L. Briant, On the chemistry of grain boundary segregation and grain boundary fracture, Metallurgical Transactions A 1990 21:9 21, 2339 (1990).
  * Messmer and Briant [1982] R. P. Messmer and C. L. Briant, The role of chemical bonding in grain boundary embrittlement, Acta Metallurgica 30, 457 (1982).
  * Krakauer and Seidman [1993] B. W. Krakauer and D. N. Seidman, Absolute atomic-scale measurements of the Gibbsian interfacial excess of solute at internal interfaces, Physical Review B 48, 6724 (1993).
  * Maugis and Hoummada [2016] P. Maugis and K. Hoummada, A methodology for the measurement of the interfacial excess of solute at a grain boundary, Scripta Materialia 120, 90 (2016).
  * Nord  _et al._ [2017] M. Nord, E. Vullum, I. Maclaren, T. Tybell, and R. Holmestad, Atomap: a new software tool for the automated analysis of atomic resolution images using two-dimensional gaussian fitting, Adv Struct Chem Imag 3, 9 (2017).
  * Malis  _et al._ [1988] T. Malis, S. C. Cheng, and R. F. Egerton, Eels log-ratio technique for specimen-thickness measurement in the tem, Journal of Electron Microscopy Technique 8, 193 (1988).
  * Madsen and Susi [2021] J. Madsen and T. Susi, The abtem code: transmission electron microscopy from first principles, Open Research Europe 1, 13015 (2021).