Secondary Grain Boundary Dislocations Alter
Segregation Energy Spectra

Xinren Chen, William Gonçalves, Yi Hu, Yipeng Gao, Patrick Harrison, Gerhard Dehm, Baptiste Gault, Wolfgang Ludwig, Edgar Rauch, Xuyang Zhou, et al.

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To cite this version:

Xinren Chen, William Gonçalves, Yi Hu, Yipeng Gao, Patrick Harrison, et al. Secondary Grain Boundary Dislocations Alter Segregation Energy Spectra. Nature Communications, 2025, 16 (1), pp.8422. 10.1038/s41467-025-64265-6 . hal-04922151

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Secondary Grain Boundary Dislocations Alter
Segregation Energy Spectra

Xinren Chen<fnref n="1" />, William Gonçalves<fnref n="2" />, Yi Hu<fnref n="1" />, Yipeng Gao<fnref n="1" />,
Patrick Harrison<fnref n="3" />, Saurabh Mohan Das<fnref n="1" />, Gerhard Dehm<fnref n="1" />, Baptiste Gault<fnref n="1" />,<fnref n="4" />,
Wolfgang Ludwig<fnref n="2" />, Edgar Rauch<fnref n="3" />, Xuyang Zhou<fnref n="1" />*, Dierk Raabe<fnref n="1" />

Grain boundaries (GBs) trigger structure-specific chemical segregation of so-
lute atoms. According to the three-dimensional (3D) topology of grains, GBs
— although defined as planar defects — cannot be free of curvature. This
implies formation of topologically-necessary arrays of secondary GB disloca-
tions. We report here that these pattern-forming secondary GB dislocations
can have an additional and, in some cases, even a much stronger effect on GB
segregation than defect-free GBs. Using nanoscale correlative tomography
combining crystallography and chemical analysis, we quantified the relation-
ship between secondary GB dislocations and their segregation energy spectra
for a model Fe-W alloy. This discovery unlocks new design opportunities for
advanced materials, leveraging the additional degrees of freedom provided by
topologically-necessary secondary GB dislocations to modulate segregation.

<footnote n="1">¹Max-Planck-Institut for Sustainable Materials, Düsseldorf 40237, Germany</footnote>

<footnote n="2">²Université Lyon I, MATEIS, INSA Lyon, CNRS UMR 5510, Villeurbanne 69621, France</footnote>

<footnote n="3">³Université Grenoble Alpes, CNRS, Grenoble INP, SIMaP, Grenoble F-38000, France</footnote>

<footnote n="4">⁴Department of Materials, Royal School of Mines, Imperial College, London, UK</footnote>

<footnote>*To whom correspondence should be addressed; E-mail: x.zhou@mpie.de</footnote>
Metals and alloys usually consist of crystalline grains that fill three-dimensional (3D) space. These grains meet at junctions to form piecewise planar defects known as grain boundaries (GBs). Due to topological reasons, GBs of real polycrystalline materials exhibit curvature (6), to reconcile force equilibrium at nodes and junctions with pore-free, space-filling topology of the non-platonic shaped grains (2). These curvature variations are realized by the build-up of disconnections, including steps and secondary GB dislocations (3, 23, 33, 42). Secondary GB dislocations, which are GB line defects with dislocation character and often form patterns with regular spacing (see Fig. 1A-C), are well known to play particularly important roles in GB migration (20, 40), sliding (61), and rotation (53).

Here we show that secondary GB dislocations also alter GB segregation. Impurities or solutes tend to segregate to GBs to minimize their total free energy, as originally described by the Gibbs adsorption isotherm (30, 31, 38, 65). Due to their localized strain fields, the pattern-forming secondary GB dislocations can induce additional — and sometimes even much stronger — GB segregation compared to the segregation in defect-free GB structures. More specific in the model system FeW studied here we find that this effect can result in a twofold increase in the composition of GB segregation compared to a defect-free GB segment. Interactions between secondary GB dislocations and solute atoms significantly alter the mechanical and physical properties of materials (71). Capturing these complex solute-GB interactions at highest resolution scale is critical for understanding GB decoration and unlocking atomic-scale engineering of alloys (41).

There are four major challenges for the experimental characterization of solute-GB interactions. Firstly, GBs often deviate from flat planes and exhibit undulating 3D structures (62, 66). Real GB structures invariably break down into facets and regular patterns formed by secondary GB dislocations, to accommodate topological constraints (2, 6). These curved surfaces contain defects that significantly alter solute segregation behavior and create local solute enrichment
patterns (23, 29, 39, 64, 67–69). To capture the full structural and chemical characteristics of a GB, one must conduct these measurements in 3D. Accurate quantification of solute segregation requires atomic-scale spatial resolution and high chemical sensitivity, as solute segregation is confined, typically to only a few atomic layers from the GB interface (25, 30, 38, 50, 63, 65). In addition, a complete representation for any GB requires the measurement of five kinematic degrees of freedom (18): three to describe the misorientation between adjacent grains and two to define the orientation of the GB segment plane. Finally, different types of GBs exhibit different segregation behaviors, sometimes varying by more than an order of magnitude. To conduct a statistically relevant analysis, it is thus essential to have the capability to study multiple GBs with a range of misorientations and GB plane orientations, if possible, simultaneously for high throughput.

Here, we apply four-dimensional scanning transmission electron microscopy (4DSTEM) tomography (19) to gain structural insights and use atom probe tomography (APT) for compositional analysis on a model body-centered cubic (BCC) Fe-1 at.% W alloy. This correlative nanoscale tomography approach enables the measurement of five kinematic degrees of freedom for any GB (18) and quantifies near atomic-scale solute-GB interactions at defect-rich GBs. 4DSTEM tomography efficiently detects defect contrasts at GBs, eliminating the need for meticulous crystal tilting required by conventional transmission electron microscopy (TEM) and simplifying the characterization of randomly oriented GBs with undulating 3D structures. By thoroughly examining both the structural and chemical characteristics across 12 Fe GBs, we provide quantitative analyses for revealing the correlations between GB structure and W segregation. This knowledge is essential to quantify the interactions between secondary GB dislocations—necessitated by curvature variations—and alloying elements, which opens design opportunities for engineering future materials.
Correlative tomography for three-dimensional crystallography and chemistry

We prepared the model BCC Fe-1 at.% W alloy thin film using physical vapor deposition (PVD) accompanied by a heat treatment at 500 °C for 240 minutes to activate diffusion and facilitate solute segregation at GBs and the secondary defects on them. The average grain size is approximately 134 nm, with equiaxed grains and a weak {111} fiber texture in the growth direction (see fig. S1). Periodic intensity contrasts in the weak-beam dark-field TEM images of high-angle GBs (HAGBs) typically stem from secondary GB dislocations, particularly in those nearly matching the coincidence site lattice (CSL) relationship (15). Figure 1D-F present the bright-field and weak-beam dark-field images of a Σ11 GB that has a 2.2° deviation from the theoretical misorientation. We confirmed that secondary GB dislocations are the source of the periodic intensity contrast, observable under two-beam conditions with g = [130] (see Fig. 1E).

To further elucidate the interactions between solutes and defects (specifically secondary GB
dislocations), we employed our recently introduced nanoscale tomography approach (18, 19).
Figure 1G-I shows a 3D crystallographic reconstruction of 11 grains and 12 GBs in a needle-
shaped Fe-W specimen, providing a meaningful reference for the GB segregation landscape
in materials with defect-containing GBs. We colored each grain based on its Euler angle
representation of the crystallographic orientation relative to the Z-axis (see table S1), along
which the specimen is tilted for tomography (see Fig. 1G), with the X-axis aligned parallel to
the thin film growth direction. The orientation was determined by analysis of the nanobeam
diffraction patterns that were systematically collected during the 4DSTEM scans. Figure 1H
displays three example diffraction results taken at different tilts. Our method enables the mapping
of local normal to the GB plane between adjacent grains, as demonstrated in fig. S2. Details on
the sample preparation and data processing are available in the Supplementary Materials Section
Sample preparation, Section 4DSTEM tomography, fig. S3A, and video.S1.
<figure><img src="image_3.png" /><figcaption>**Fig. 1.** Correlative tomography characterization of the Fe-1 at.% W specimen. (A) Illustration of grain boundaries (GBs) in a three-dimensional (3D) polycrystalline structure. (B) Magnification of a selected high-angle GB (HAGB) showing curved GB plane. (C) Illustration of steps and secondary GB dislocations on the GB. (D) Bright-field image of a transmission electron microscopy (TEM) lamella prepared from the body-centered cubic (BCC) Fe-1 at.% W specimen, featuring multiple GBs, one of which is labeled as a HAGB. This HAGB is a Σ11 GB that deviates by 2.2° from the theoretical misorientation. (E) Corresponding weak-beam dark-field image of the same area, imaged under the two-beam condition with g = [1¯30]. The inset image in the right side of (E) displays the diffraction pattern collected from the left grain. Scale bar: 1/0.1 nm. (F) High magnification bright-field image displaying the step feature of the GB. (G) Illustration of four-dimensional scanning transmission electron microscopy (4DSTEM) tomography. Each pixel in the 4DSTEM data incorporates a local nanobeam diffraction pattern. (H) Representative nanobeam diffraction patterns from the 4DSTEM datasets at different tilt angles for the grain indicated by red circle in (G). Scale bar: 1/0.083 nm. (I) 3D crystallographic reconstruction of the grains in the Fe-W needle-shaped specimen, characterized via 4DSTEM tomography (18, 19). Grains are colored based on the Euler angle representation of their crystallographic orientations relative to the Z-axis, serving as the tilting axis in TEM. The X-axis represents the thin film growth direction in the same coordinate system. (J) 3D chemical reconstruction of the same specimen shown in (I) by atom probe tomography (APT), illustrating the spatial distribution of Fe (red) and W (cornflower blue) atoms with a superimposed 5 at.% W isosurface. The 5 at.% isosurface represents the region (voxels) containing 5 or more at.% W. For a clear visualization, we aligned the APT reconstruction in the same perspective as the 4DSTEM tomography reconstruction shown in (I).</figcaption></figure>
We analyzed the composition of this same specimen by APT. Figure 1J displays the spatial distribution of Fe as individual red dots, and a set of isocomposition surfaces that delineates regions containing 5 or more at.% W. The W enrichment closely correlates with the location of GBs identified in the 3D crystallographic reconstruction in Fig. 1I, evidencing preferential W segregation to GBs. We focus particularly on the following two GBs with a near-Σ relationship that exhibit discontinuous W segregation patterns.

**Quantitative analysis of secondary grain boundary dislocations and chemical segregation**

To quantify segregation patterns, a 3D compositional mapping of W within the APT reconstruc-
tion was calculated with a resolution of 0.5 × 0.5 × 0.5 nm³ and a delocalization parameter of
3 nm (21). The compositional map plotted in Fig. 2A shows that the segregation of W at GBs
forms a discontinuous pattern rather than a uniform planar distribution. We selected two GBs
for detailed analysis: GB1 between grain α₁ and grain α₂, characterized as a near Σ5[100] GB
with a deviation of 5.6° from its ideal misorientation, and GB2 between grain α₄ and grain α₅,
identified as a near Σ13b[111] GB with a 3.4° deviation, illustrated in Fig. 2B-E and Fig. 2F-I,
respectively.

Figure 2B presents two mappings of the local normal to the GB plane between adjacent
grains for GB1: the bottom mapping corresponds to grain α₁, and the top to grain α₂. The
similarity in color representation across both mappings indicates that the orientations of the GB
planes on both sides of the interface are similar, suggesting a near-symmetric tilt Σ5[100] GB.
We identified secondary GB dislocations by constructing a virtual dark-field image based on
the vector **g** = [121] for grain α₂, as shown in Fig. 2C. See more details in fig. S4. Figure 2D
displays a stairwell-like periodic segregation pattern at this GB, highlighted by a set of isosurfaces
with a threshold of 2.5 at.% W. A one-dimensional (1D) composition profile with a bin size of
1 nm was calculated along the cylinder positioned through the segregation pattern in Fig. 2D.
From this profile, plotted in Fig. 2E, we quantify the level of segregation ranging from 1 to 4 at.% W. Note that this pattern exhibits a regular spacing of approximately 22 nm, a spacing that closely matches the spatial distribution of secondary GB dislocations imaged by virtual dark-field in Fig. 2C. The maximum value in the local composition profile is twice that of the mean composition of the GB, which represents that of a defect-free GB segment. This direct correlation suggests a key role of secondary GB dislocations in altering solute segregation, in terms of mechanism, trapping depth, patterning, magnitude and kinetics.

<figure><img src="image_4.png" /><figcaption>**Fig. 2. Characterization of secondary GB dislocations and their linkage to segregation patterns in the Fe-1 at.% W specimen.** (A) Compositional mapping of W at a resolution of 0.5 × 0.5 × 0.5 nm³ per voxel, enabling visualization of volume-specific variations. Parts of the grains and GBs have been labeled, with identities ranging from grain α₁ to grain α₅ and from GB1 to GB2 (see all labels in fig. S2). Correlative crystallographic and compositional quantitative analysis for two GBs (GB1: B-E) and (GB2: F-I). Each set includes: (B & F) orientation mappings of the local normal to the GB plane between adjacent grains (rendered in translucent grey), with two mappings provided for each GB, referenced to the respective grain involved; (C & G) 4DSTEM virtual dark-field images; (D & H) atom maps of Fe (colored red) and W (colored cornflower blue), with superimposed isosurfaces at 2.5 at.% W and 4.0 at.% W, respectively; (E & I) W profiles along the red arrows shown in (E) & (I). The insert images in (C) & (G) display the nanobeam diffraction patterns with the vectors $g = [12\bar{1}]$ for grain α₂ and $g = [2\bar{1}11]$ for grain α₅ (highlighted by red circles), which were used to generate the virtual dark-</figcaption></figure>
field images shown in (C, G). Two perspectives (90° rotated clockwise from left to right) in (D & H) are shown to illustrate the segregation patterns in 3D.

We observe similar results at the Σ13b[1 1 1] GB, as shown in Fig. 2F-I. Unlike the previous case, GB2 is an asymmetric tilt GB, with the plane normal for grain α₄ aligning with (111̄2) and for grain α₅ with (110̄), as detailed in Fig. 2F. Nevertheless, we still find a close match between the secondary GB dislocations shown in Fig. 2G and fig. S5, and the periodic segregation patterns (see Fig. 2H) and the composition profile plotted in Fig. 2I. We also observed GB facets that can modulate solute segregation at GBs, similar to the reports in references (29, 32, 67, 68). These were particularly noted at GB3 (see fig. S6).

Calculation of grain boundary segregation energy using the Langmuir-McLean isotherm

The relationship between solute segregation and segregation energy typically follows the Langmuir-McLean isotherm, as detailed in Equation (1) (26, 31, 35, 59),

$$ \frac{X_{GB}}{1 - X_{GB}} = \frac{X_{B}}{1 - X_{B}} \exp \left( \frac{\Delta G_{Seg}}{RT} \right). \qquad (1) $$

Here, ΔG<sub>Seg</sub> is the segregation energy of the solute atom, R is the ideal gas constant, T is the temperature, and X<sub>B</sub> and X<sub>GB</sub> are the compositions of solute in the bulk and at the GB, respectively. X<sub>GB</sub> can be further expressed as (Γ·Ω(1-X<sub>B</sub>)/t) + X<sub>B</sub> (55), where Γ represents the interfacial excess (IE), denoting the excess number of atoms per unit area at an interface, t is the thickness of the GB, and Ω is the atomic volume of the solvent.

We mapped the IE values across all visible GBs in the APT dataset, with Fig. 3A providing examples of the integral profiles across GB1 used for these IE calculations, and Fig. 3B showing the mapped IE values (47, 70). We chose IE as it minimizes the influence of local magnification caused by different field evaporation behaviors arising from different compositions and grain orientations (34, 70). Solute segregation at GBs typically spans only a few atomic layers, which
is often smaller than the volume increment, potentially leading to inaccurate measurements of
local segregation. Consequently, IE is employed for accurate quantification analysis.

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Mapping and analysis of GB properties in the Fe-1 at.% W specimen: (A) Integral profiles across GB1 for quantifying the interfacial excess (IE) of W segregation at the Fe GB (47, 70). We plot measurements from two points: Point 1 for high W regions (colored pink) and Point 2 for low W segregation regions (colored orange). In each plot, the solid lines show the cumulative relationship between all atoms and solute atoms, while the dashed lines are the fittings within the two grains adjacent to the GB plane. $N_W^{excess}$ represents the accumulation of excess atoms across the GB interface. IE values are calculated by dividing $N_W^{excess}$ by the corresponding interface areas, which are approximately 8.0 nm$^2$ for Point 1 and Point 2. We indicate the locations of Point 1 and Point 2 in the embedded image, consistent with Fig. 2D. (B) IE mapping on GBs, as identified from the correlative characterization in Fig. 1I & J. We labeled the locations of all investigated GBs from GB1 to GB12 in (B). (C) Segregation energy mapping on the same GBs as in (B), calculated based on the Langmuir-McLean isotherm, Eq. (1) (26, 31).</figcaption></figure>
The locations where 1D line profiles of segregation energy values were measured are marked by red lines S1 and S2, details of which will be presented in the following section. (D) Correlation between the misorientation angle and segregation energy of all investigated GBs: This scatter plot displays the average segregation energy (in J/mol) for various GBs, plotted against their misorientation (in degrees). Each point represents a different GB (GB1 to GB12), with colors indicating the rotation axis of misorientation related to the cubic crystal symmetry (*m*3*m*), as shown in the color triangle legend. Experimental GB segregation energy spectra for (E) GB1, (F) GB2, and (G) all GBs. The fitting plots, overlaid onto the histograms, generally follow the skew-normal function (59) but with local deviations highlighted by the black arrows, see (E & F). The used fitting parameters are: characteristic energy (μ) in kJ/mol, width (σ) in kJ/mol, and shape parameter (α) along with each plot. The sharp peak appearing near 0 J/mol in panel (G) results from the contribution of the GBs without solute segregation and the inevitable minor inclusion of the bulk regions adjacent to the GBs.

Figure 3C maps the segregation energy calculated from the IE, based on the Langmuir-McLean isotherm, Eq. (1). While this isotherm does not account for solute-solute interactions, for more complex cases, models such as the Fowler-Guggenheim isotherm (12) or the Seah-Hondros isotherm (46) could be similarly applied by adding an extra term to account for the interaction energy. Despite these complexities, we hypothesize that the general formulation suggested by the Langmuir-McLean isotherm (Eq. (1)) remains applicable here, except for the case that it cannot account for any second order defect trapping on the GB, such as caused by GB dislocation structures.

We calculated the mean segregation energy for each GB. As plotted in Fig. 3D, it shows the same trends previously reported from low angles to 60° (22), demonstrating a clear effect of misorientation on the segregation level. Our measurements allow for much more refined analysis of the segregation energy landscape. We have generated segregation energy spectra, Fig. 3E-G for GB1, GB2, and all 12 GBs. The statistical distribution of solute segregation is generally close to a skew-normal function (see Fig. 3E & F for GB1 and B2), as predicted by atomistic simulations (59), a feature that had not yet been confirmed before by experiment. The experimental segregation energy spectra of defect-containing GBs deviate from the ideal curve,
as marked by black arrows in Fig. 3E & F.

## Estimation of the elastic energy contribution for grain boundaries containing secondary dislocations

To identify the impact of secondary GB dislocations on solute segregation, we analyzed the periodic variations in segregation energy along GB1 and GB2 (see Fig. 3C, Lines S1 and S2), using 1D composition profiles plotted in Fig. 4A & B. The gradient in these profiles can be attributed to the secondary GB dislocations visible in Fig. 2C & G. The magnitude of the segregation energy change caused by these secondary GB dislocations can reach up to approximately 6000 J/mol (corresponds to 62 meV/atom).

Secondary GB dislocations generate a stress field that is nonlinear and contains non-elastic contributions within the dislocation core region, and is linear and elastic outside of the core. The dislocation core, which normally spans the length of two Burgers vectors (27), serves as an essential trap for solutes. Notably, the core size is smaller than the binning scale used for our segregation energy quantification, which is approximately 2 nm (see the line profile in Fig. 4A & B). Our main goal in this paragraph is to understand the gradient in these composition profiles. Thus, our analysis only focuses on the linear and elastic part of the stress field associated with the secondary GB dislocations. The significant impact of secondary GB dislocations on solute segregation necessitates the introduction of an additional term to accurately represent the change in segregation energy. The total segregation energy at the GB is expressed as:

$$ \Delta G_{\text{seg}} = \Delta G_{\text{seg}}^{\text{in}} - P^{\text{XS}}\Delta V - T\Delta S_{\text{seg}}^{\text{XS}}. \quad (2) $$

Here, $\Delta G_{\text{seg}}^{\text{in}}$ represents the intrinsic segregation energy of the solute at the GB, excluding contributions from secondary GB dislocations and accounting for the sum of the segregation enthalpy, elastic energy, and segregation entropy (17), $P^{\text{XS}}$ denotes the hydrostatic pressure caused by the presence of secondary GB dislocations, $\Delta V$ represents the change in volume
required for solute atoms to replace solvent atoms, and $\Delta S_{seg}^{XS}$ refers to the change in excess segregation entropy due to the introduction of secondary GB dislocations. $\Delta G_{seg}^{in}$ depends on the kinematic degrees of freedom for a given GB (10, 60), while $\Delta S_{seg}^{XS}$ reflects the change in the configurations due to the presence of secondary GB dislocations. We neglect the $\Delta S_{seg}^{XS}$ term and primarily attribute the modulation of segregation energy within a GB to the elastic energy resulting from dislocations at the GBs (see Fig. 2C & G).

For bulk dislocations, the surrounding stress field can induce solute segregation in the areas surrounding the dislocations, known as Cottrell atmospheres (4, 9, 49, 52). In the case of secondary GB dislocations, the same applies, leading to a similar solute segregation phenomenon. Unlike Burgers vectors in the bulk lattice, the unit Burgers vectors at GBs (23) are known as the displacement shift complete (DSC) vectors, shown in table S2 for GB2. The pronounced contrast in secondary GB dislocations, seen in Fig. 2C & G, suggests the formation of GB ledges through the agglomeration of individual secondary GB dislocations, as documented in previous studies (36, 37).

We chose GB2 between grain $\alpha_4$ and grain $\alpha_5$ ($\Sigma13b$) to calculate the elastic field around secondary GB dislocations (see fig. S7 and fig. S8) using the Stroh formalism for anisotropic elasticity theory (48, 54, 57) (see Methods Stress field simulation). The Stroh formalism requires a cutoff distance near the dislocation core, beyond which linear anisotropic elasticity theory applies and within which it fails. Here, a cutoff of 0.5 nm was used for the stress field simulations. It is worth mentioning that non-linear anisotropic elasticity calculations by Lazar et al. (27) and atomistic simulations by Clouet et al. (7) are theoretical methods for estimating the stress field in the dislocation core. Inputs for the stress field calculations were obtained directly from our 4DSTEM tomography analysis, i.e., the five kinematic degrees of freedom for GB2 (see fig. S2A & B and table S3). Figure 4C & D show a periodic pattern in the stress field around the secondary GB dislocations with the DSC-lattice vectors $\mathbf{b}_{\alpha5} = \frac{a}{13}[\bar{1}4\bar{3}]$ and $\mathbf{b}_{\bar{3}5} = \frac{a}{13}[\bar{3}\bar{1}4]$ function as the
Burgers vectors (24). Here, *a* is the lattice constant of BCC Fe.

<figure><img src="image_6.png" /><figcaption>**Fig. 4.** Quantifying elastic energy for the Σ13b GB with dislocations using linear anisotropic elasticity theory. (A & B) Segregation energy profiles along the lines marked by red arrows in Fig. 3C: (A) Line S1 at GB1 and (B) Line S2 at GB2, illustrating variations in segregation energy along these GBs. Horizontal dashed lines in (A) and (B) indicate the mean segregation energy of GB1 and GB2, respectively, serving as references. (C & D) The stress field surrounds secondary GB dislocations for GB2 (Σ13b, see Fig. 2F-I) with the displacement shift complete (DSC)-lattice vectors as Burgers vectors: (C) DSC-a: $b_{α5} = \frac{a}{13}[\bar{1}4\bar{3}]$ and (D) DSC-b: $b_{α5} = \frac{a}{13}[\bar{3}1\bar{4}]$. Here, *a* is the lattice constant of BCC Fe. (E) The change in segregation energy ($P^{XS}\Delta V$) aligns along the GB, corresponding to (C & D). Here, DSC-a and DSC-b have the same Burgers vectors as those in (C & D). The modulation factor is defined as $\exp(-\frac{P^{XS}\Delta V}{RT})$.</figcaption></figure>

The orientation of the Burgers vector significantly influences the elastic field around the GB cores by varying its intensity and spread (see Fig. 4C & D). With the formula, $P^{XS}\Delta V$, we further calculated the extra contribution to the segregation energy from the elastic field produced by secondary GB dislocations, as presented in Fig. 4E for the quantitative measurements along the GB plane. The elastic energy contributes a modulation factor ranging from 0.4 to 2.5 times relative to the prediction of the Langmuir-McLean isotherm, corresponding to attenuation or
enhancement depending on whether it takes a value below or above unity, respectively.

Our experiments confirm that the formation of secondary GB dislocations impacts W segregation at GBs in BCC Fe, resulting in an up to 100% increase in the composition of GB segregation compared to a defect-free GB segment. Such secondary GB dislocations form patterns with regular spacing of approximately 22 nm, which are topologically necessary to accommodate the GB's local curvature. We have successfully quantified the segregation energy spectra of these GBs using interfacial excess mapping from APT data. The deviation in segregation energy within these spectra can reach magnitudes of up to approximately 6000 J/mol. The extra elastic energy caused by secondary GB dislocations introduces a modulation factor, for example, ranging from 0.4 to 2.5 for the Σ13b GB, altering the local composition relative to predictions based on the Langmuir-McLean isotherm. These findings underscore the critical role of secondary GB dislocations not just as a topological necessity to accommodate GB shapes but also as deep solute traps, massively modulating solute segregation by up to a factor of two. This enhances our understanding of GB decoration, offering new opportunities for the design of advanced alloys.

## Acknowledgments

X.Z., W.L., and E.R. acknowledge funding by the German Research Foundation (DFG) through the project HE 7225/11-1. X.C. and B.G. gratefully acknowledge the Collaborative Research Centre/Transregio (CRC/TRR) 270 HoMMage-Z01 project funded by the DFG. The authors acknowledge Prof. Dr. Christian Liebscher for the discussion on transmission electron microscopy results.
**Supplementary materials**

Materials and Methods

Supplementary Text

Figs. S1 to S10

Tables S1 to S3

Videos S1 to S3

References (28-71)
# Materials and Methods

## Sample preparation

We deposited the model alloy Fe-1at.%W thin film sample in a PVD cluster (BESTEC GmbH, Berlin, Germany). The synthesis process involved co-sputtering a pure Fe target (99.995%, Mateck, Germany) in a direct current cathode with a power of 130 W, and a W target (99.95%, Kurt J. Lesker, USA) in a radio frequency cathode at 28 W, resulting in a total thickness of approximately 2000 nm. Prior to sputtering, the chamber was pumped to a base pressure of $4.0 \times 10^{-8}$ mbar. The Fe-W alloy thin films were then deposited at a pressure of $5.0 \times 10^{-3}$ mbar and an Ar flux of 40 sccm on smooth substrates of single crystalline silicon [100] wafers with a 1.5 µm thermal $SiO_2$ diffusion and reaction barrier layer (see fig. S1A), which was placed 110 nm away from the sputtering targets. The specimens were held at 500 °C for 240 minutes to facilitate solute segregation at GBs.

We prepared the needle-shaped specimen for correlative 4DSTEM tomography and APT characterization using the lift-out and annular milling technique developed by Thompson et al. (11, 51). The preparation of the specimen was conducted in a plasma-FIB (PFIB) instrument (FEI Helios PFIB) equipped with an Xe-ion source, which results in a low penetration depth that minimizes ion implantation and amorphization in the specimen. The lift-out wedges were directly mounted onto an APT-compatible cylindrical Cu post, ready for ± 90° rotation along the tilting axis in the Fischione Model 2050 On-Axis Rotation Tomography Holder in the TEM. During the lift-out procedure, we carefully aligned the rotation axis of the sample with that of the Cu post to match the tilt axis of the tomography holder. A final PFIB milling condition of 10 pA at 5 keV was applied to shape the needle-shaped specimen to a top diameter of 100 nm (see fig. S1E), while ensuring a clean surface.
## 4DSTEM tomography

We conducted the 4DSTEM tomography characterization using a JEM-2200FS (JEOL Ltd.) microscope, which was operated at an accelerating voltage of 200 kV and equipped with a 10 µm condenser (CL1) aperture and a 50 eV energy filter. This 4DSTEM tomography method relies on virtual dark-field images reconstructed from 4DSTEM datasets acquired at different tilts (18, 19). A schematic of the 4DSTEM tomography characterization techniques is presented in fig. S3A. We collected 17 4DSTEM scans at tilts ranging from -80° to +80° with a tilt step increment of 10°. During the scan, a quasi-parallel electron probe, approximately 2 nm in diameter, traverses a selected area of 100 nm × 300 nm in a line-by-line manner with a 1 nm step, capturing nanobeam diffraction patterns at each probe position using a 4k × 4k CMOS detector (TemCam-XF416-TVIPS) with a pixel dwell time of 41 ms.

The phase and orientation of the grains in the 4DSTEM datasets were determined automatically using the multi-index algorithm, which retrieves information about overlapping grains via the ASTAR software package (43, 44) (see fig. S2). This crystallographic information serves as a guideline for 1) precisely refining the tilting angles among different tilts; and 2) automatically generating virtual dark-field images for each grain in every 4DSTEM scan using the frozen template algorithm (18, 19, 43). We conducted the coarse alignment of the virtual dark-field images manually in Tomviz (28), followed by fine alignment using the PyCorrectedEmissionCT (corrct) package (58). These well-aligned virtual dark-field images of each grain were used for 3D tomographic reconstruction through the Simultaneous Iterative Reconstruction Technique (SIRT) algorithm, which included 15 iterations combined with a non-negative minimum constraint to promote physical solutions. For accurate determination of GB locations in volumes of overlapping grains, a max-pooling type approach is applied, which assigns the highest intensity volumes to the corresponding grains. We then employed the marching cubes algorithm to generate normals for the planes corresponding to the two grains adjacent to each GB, thus providing
distinct plane definitions for both (56). We used Blender (8) to animate the method and visualize
the reconstructed volume, as demonstrated in Videos s1-s3.

Correlative atom probe tomography

We analyzed the chemistry of the same needle-shaped specimen using APT after acquiring
the 4DSTEM datasets. Prior to loading the specimen into the APT analysis chamber, it was
cleaned with low-kV Ar ions using a Gatan PECS Model 682 system at 2 kV and 32 µA to
remove hydrocarbon layers accumulated during the TEM measurements. We then conducted
the APT measurements using a LEAP 5000XS instrument (Cameca Instruments) operated with
a specimen temperature of 70 K and a laser pulse energy of 100 pJ at a pulse repetition rate of
200 kHz for a 2% ions per pulse detection rate. As shown in Fig. 1J and fig. S3B, we successfully
collected 300 million ions from this correlative specimen, a critical volume that enables the
simultaneous investigation of multiple GBs. Data reconstruction was performed using the AP
Suite 6.3 software package, following a calibration procedure to achieve the correct image
compression factor and k-factor, which are essential for accurately shaping and spacing the
lattice in the reconstructed volume (14).

We employed the APT_GB software (70) to analyze the in-plane chemical distribution of W
atoms at the GBs. With the pre-trained convolutional neural network, we automatically identified
the locations of GB planes from the APT dataset, which were represented as triangular meshes
with an average unit size of approximately 7 nm². We generated integral profiles across the GBs
to quantify the IE at each vertex on the mesh, with examples shown in fig. S9). The resulting IE
map is shown in Fig. 3B.

Displacement shift complete lattice of the Σ13b grain boundary

The dichromatic pattern of the DSC lattice of the Σ13b GB (see fig. S10) and the analysis of
Burgers vectors for secondary GB dislocations were obtained using the DSC lattice through
O-lattice theory (5), utilizing an in-house developed open-source GB code (16).

## Stress field simulation

Misfit dislocations on GBs can induce a stress field that would potentially change the segregation behavior of solute atoms in the alloy system. Such stress field was studied in previous work (1,57), where a Stroh formalism is used to obtain the corresponding periodic stress field generated by semicoherent boundary and the required misfit dislocations. In essence, we solve the mechanical equilibrium equation written as

$$ \sigma_{ij,j}(x_1, x_2) = C_{ijkl} u_{k,jl}(x_1, x_2) = 0 \quad (3) $$

in both upper and lower grain with the critical interface condition expressed as

$$ \begin{aligned} [[\sigma_{2k}(x_1, x_2 = 0)]] &= 0 \\ [[u_k(x_1, x_2 = 0)]] &= -\sum_{n=1}^{\infty} \frac{1}{\pi n} \sin\left(\frac{2\pi n}{d}x_1\right) b_k \end{aligned} \quad (4) $$

where [[f]] denotes the jump function across the interface (interface normal is $x_2$ axis), d is the dislocation spacing, $b_k$ is dislocation Burgers vector. This condition ensures that traction is in equilibrium across the interface and the relative displacement is consistent with our interfacial dislocation pattern.

To solve the equilibrium (3) on a periodic domain, Fourier series are used to write displacement field.

$$ u_k(x_1, x_2) = 2\mathrm{Re} \sum_{n=1}^{+\infty} e^{i \frac{2\pi n}{d} x_1} \tilde{u}_k(x_2) \quad (5) $$

where $\tilde{u}_k(x_2)$ is the Fourier coefficient depending on position $x_2$. Substituting the displacement field into mechanical equilibrium together with further manipulation, the system boils down to a sextic equation which is the same with the one in a common Stroh formalism. To satisfy interfacial boundary condition, six complex scaling parameters can be assumed, which will
then transform (4) to a set of six independent linear equation. Once the six complex scaling
parameters are solved, the stress field can be obtained by calculating the strain with the solved
displacement field and multiplying with the elastic constant in each grain (see fig. S7 and fig. S8).

Area size: $L = 48 \times 48 \text{ nm}^2$; Number of grid points: $500 \times 500$; Upper grain Euler angles: $[41, 54.7, 45]$; Lower grain Euler angles: $[8, 54.7, 45]$; Elastic constants of BCC Fe (13): $C_{11} = 257.7 \times 10^9 \text{ Pa}$, $C_{12} = 144.0 \times 10^9 \text{ Pa}$, $C_{44} = 94.9 \times 10^9 \text{ Pa}$; Poisson's ratio: $\nu = 0.3$; Burgers vector of the dislocation: $b = 0.083 \text{ nm}$; The core size of dislocations is defined as 2 Burgers vectors.
Supplementary Figures

<figure><img src="image_7.png" /><figcaption>**Fig. S1. Typical microstructure of the as-prepared Fe-1 at.% W thin film.** (A) Definition of the coordination system used to describe the geometry of the Fe-1 at.% W thin film specimen. Here, the X-axis is aligned parallel to the thin film growth direction. The Y and Z-axes, arbitrarily selected, lie in the plane perpendicular to the thin film growth direction. Once the needle-shaped specimen is prepared, the Z-axis serves as the tilting axis in TEM. (B) Region and (C) the corresponding orientation map for the transmission Kikuchi diffraction (TKD) analysis of the thin film prepared normal to the X-axis, termed as the plane view specimen. (D) (1 1 1) pole figure, oriented parallel to the thin film growth direction, for the region shown in (C). (E) Bright field image of the as-prepared Fe-1 at.% W APT sample.</figcaption></figure>
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>**Fig. S2.** 3D crystallographic reconstruction of grains and GBs in the correlative Fe-1at.% W needle-shaped specimen. (A) The orientations of grains and (B) the crystallographic information of GBs, both from the same sample as shown in Fig. 1I. In (B), the local normal to the GB plane has been added, and grains have been manually detached to enhance clarity. Each GB features two plane normal mappings between adjacent grains. The coordinate system for each GB plane normal mapping is referenced to the respective grain involved in the mapping.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>**Fig. S3.** Schematics of the correlative tomography characterization techniques. (A) View of the internal optics, not to scale, inside a TEM column optimized for 4DSTEM data collection. The process involves collecting multiple 4DSTEM datasets at various tilts to generate a series of virtual dark-field images for all grains using the frozen template algorithm (18, 19). Employing our in-house developed SPED3D software (18, 19), we achieved 3D reconstructions, with the phases and orientations of all grains quantified. (B) View of the internal setup, not to scale, inside an APT analysis chamber configured with a straight flight-path. Following TEM analysis, the specimen is transferred to the chamber and undergoes field evaporation ion by ion, with the coordinates and mass-to-charge ratio of each ion recorded for detailed 3D reconstruction and chemical analysis. X₀, X₁, Y₀ and Y₁, times obtained from the delay-line detector used to determine the ion impact positions.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Fig. S4. Dislocation analysis for the Fe-1 at.% W specimen.** (A) Displays the overlaid nanobeam diffraction patterns of grains α₂ to α₄. (B) Shows a reconstructed 4DSTEM virtual dark-field image created using the virtual aperture depicted in (A), which includes potential diffraction spots calculated from grains α₂ to α₄. The 4DSTEM dataset captures nanobeam diffraction patterns at each scanning pixel, allowing for the use of a virtual aperture to select one or multiple spots from the recorded patterns to reconstruct virtual dark-field images (43). The aperture set shown in (A) includes the g = [121] spot of grain α₂ (orange circle), the g = [011] spot of grain α₃ (red circle), and the g = [312] spot of grain α₄ (green circle), effectively highlighting grains α₂, α₃, and α₄. The diffraction pattern in (A) for grain α₃ illustrates a close two-beam condition. The corresponding virtual dark-field images for (011) reveal the contrast of dislocations, with Burgers vectors satisfying b·g ≠ 0 (45). We further analyze the line contrast on grain α₄ in (B) using the b·g ≠ 0 criterion, confirming that the line contrasts are due to dislocations.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Fig. S5.** Virtual dark-field reconstruction from the 4DSTEM dataset for grain α₅. (A) Diffraction pattern of grain α₅. (B) Indexing of the diffraction pattern shown in (A). (C) Corresponding 4DSTEM dark-field images of the virtual aperture shown in (A). (D) Template-aperture set based on the crystal orientation of grain α₅. (E) Template-aperture dark-field images.</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Fig. S6. Characterization of GB facets and their link to segregation patterns in the Fe-1 at.% W specimen.** (A) The same mapping as in Fig. 2A. (B) The isosurfaces superimposed at 8.0 at.% W on the atom maps of Fe and W highlight the GB indicated by the red arrow in (A). (C) Mapping of the local normal to the GB plane between adjacent grains, from the grain α₃ side and the grain α₄ side, respectively. (D) Side view of the GB within the blue dashed-line frame shown in (B), illustrating a staircase GB segregation pattern and indicating GB facets. The superimposed arrays of atoms represent the crystalline lattices for grains α₃ and α₄. (E) Compositional profile of W along the red arrow in (D). The numbers ① and ② in (C-E) indicate the same positions across different plots.</figcaption></figure>
<figure><img src="image_16.png" /></figure>

<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /><figcaption>Fig. S7. Stress component distribution analysis for the Σ13b GB with a DSC Burgers Vector of $\frac{a}{13}[\overline{14}\overline{3}]$: (A) $\sigma_{11}$, (B) $\sigma_{22}$, (C) $\sigma_{12}$, and (D) $\sigma_{33}$.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Fig. S8. Stress component distribution analysis for the Σ13b GB with a DSC Burgers Vector of $\frac{a}{13}[\bar{3}\bar{1}4]: (A) \sigma_{11}, (B) \sigma_{22}, (C) \sigma_{12}, \text{and (D) } \sigma_{33}$.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>**Fig. S9. Visualization of solute segregation by IE with integral profiles across the GBs (A)**

IE mapping on GB1 with specific areas labeled B, C, and D. (B-D) Integral profiles across GB1 illustrate the segregation profiles of solutes for selected regions B through D. These diagrams plot the fraction of solute ions against the total number of ions counted within a cylindrical-like volume across the interface, for areas B through D as shown in (A) (70). In these subfigures, blue lines represent experimental results, while orange and red dashed lines indicate theoretical fittings for adjacent two grains. Yellow circles mark the GB positions, highlighting regions of significant atom accumulation.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>**Fig. S10.** The coincident site lattice (CSL) and DSC lattice of the Σ13b GB. The CSL lattice is represented by black dots, forming a foundational grid. Overlaying this, the DSC lattice is depicted by pink lines. The red and blue circles signify atomic positions for α4 and α5, respectively. The red and blue arrows indicate the Burgers vectors of DSC-b, $\frac{a}{13}[\bar{3}\bar{1}4]$, and DSC-a, $\frac{a}{13}[\bar{1}4\bar{3}]$, respectively.</figcaption></figure>
Supplemental Tables

<table><caption>**Table S1:** Bunge Euler Angle ($\phi_1$, $\Phi$, $\phi_2$) in degrees of all grains at the tilting angle of 0° in TEM.</caption><thead><tr><th>Grain</th><th>φ<sub>1</sub></th><th>Φ</th><th>φ<sub>2</sub></th></tr></thead><tbody><tr><td>α<sub>1</sub></td><td>244.0</td><td>31.1</td><td>43.0</td></tr><tr><td>α<sub>2</sub></td><td>122.7</td><td>34.7</td><td>239.3</td></tr><tr><td>α<sub>3</sub></td><td>257.6</td><td>46.9</td><td>40.1</td></tr><tr><td>α<sub>4</sub></td><td>37.3</td><td>25.9</td><td>282.2</td></tr><tr><td>α<sub>5</sub></td><td>-26.4</td><td>15.4</td><td>324.2</td></tr><tr><td>α<sub>6</sub></td><td>273.0</td><td>26.4</td><td>17.5</td></tr><tr><td>α<sub>7</sub></td><td>-37</td><td>16.6</td><td>4.2</td></tr><tr><td>α<sub>8</sub></td><td>57.5</td><td>29.2</td><td>292.0</td></tr><tr><td>α<sub>9</sub></td><td>-24.9</td><td>27.5</td><td>336.1</td></tr><tr><td>α<sub>10</sub></td><td>-32.1</td><td>12.3</td><td>315.3</td></tr><tr><td>α<sub>11</sub></td><td>73.9</td><td>22.1</td><td>255.1</td></tr></tbody></table>
<table><caption>**Table S2:** Unit Burgers vectors of secondary GB dislocations computed for principal coincidence system of body-centered cubic (BCC) crystals</caption><thead><tr><th>Σ values</th><th>b<sub>1</sub></th><th>b<sub>2</sub></th><th>b<sub>3</sub></th></tr></thead><tbody><tr><td>5</td><td><sup>a</sup>&frasl;<sub>5</sub>[012]</td><td><sup>a</sup>&frasl;<sub>5</sub>[021]</td><td><sup>a</sup>&frasl;<sub>10</sub>[531]</td></tr><tr><td>13b</td><td><sup>a</sup>&frasl;<sub>13</sub>[134]</td><td><sup>a</sup>&frasl;<sub>13</sub>[341]</td><td><sup>a</sup>&frasl;<sub>26</sub>[391]</td></tr><tr><td>45b</td><td><sup>a</sup>&frasl;<sub>45</sub>[542]</td><td><sup>a</sup>&frasl;<sub>5</sub>[012]</td><td><sup>a</sup>&frasl;<sub>30</sub>[517]</td></tr></tbody></table>
<table><caption>**Table S3:** Misorientation axis and angle for selected CSL GBs.</caption><thead><tr><td>GB</td><td>Misorientation Axis</td><td>Misorientation Angle θ (°)</td><td>Closest CSL</td></tr></thead><tbody><tr><td>α<sub>1</sub>|α<sub>2</sub></td><td>[20 0 -3]</td><td>34.9</td><td>Σ5 [Dev:5.6]</td></tr><tr><td>α<sub>3</sub>|α<sub>4</sub></td><td>[21 10 -19]</td><td>37.1</td><td>Σ45b [Dev:1.4]</td></tr><tr><td>α<sub>4</sub>|α<sub>5</sub></td><td>[23 -34 -32]</td><td>29.7</td><td>Σ13b [Dev:3.4]</td></tr></tbody></table>
References

1. N Abdolrahim and MJ Demkowicz. Determining coherent reference states of general semicoherent interfaces. *Computational Materials Science*, 118:297-308, 2016.

2. MF Ashby. The deformation of plastically non-homogeneous materials. *The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics*, 21(170):399-424, 1970.

3. G Ho Bishop and Bo Chalmers. A coincidence—ledge—dislocation description of grain boundaries. *Scripta Metallurgica*, 2(2):133-139, 1968.

4. D Blavette, E Cadel, A Fraczkiewicz, and A Menand. Three-dimensional atomic-scale imaging of impurity segregation to line defects. *Science*, 286(5448):2317-2319, 1999.

5. W Bollmann. On the geometry of grain and phase boundaries: I. general theory. *Philosophical magazine*, 16(140):363-381, 1967.

6. Yingbin Chen, Jian Han, Hailin Deng, Guang Cao, Ze Zhang, Qi Zhu, Haofei Zhou, David J Srolovitz, and Jiangwei Wang. Revealing grain boundary kinetics in three-dimensional space. *Acta Materialia*, 268:119717, 2024.

7. Emmanuel Clouet, Lisa Ventelon, and François Willaime. Dislocation core energies and core fields from first principles. *Physical review letters*, 102(5):055502, 2009.

8. Blender Online Community. *Blender - a 3D modelling and rendering package*. Blender Foundation, Stichting Blender Foundation, Amsterdam, 2018.

9. Ao H Cottrell and B Ao Bilby. Dislocation theory of yielding and strain ageing of iron. *Proceedings of the Physical Society. Section A*, 62(1):49, 1949.

10. Jérôme Creuze, Fabienne Berthier, Robert Tétot, and Bernard Legrand. Intergranular segregation and ordering effect: A mixed Monte Carlo mean-field approach. *Physical Review B*, 62(4):2813, 2000.

11. Gerald Da Costa, Celia Castro, Antoine Normand, Charly Vaudolon, Aidar Zakirov, Juan Macchi, Mohammed Ilhami, Kaveh Edalati, François Vurpillot, and Williams Lefebvre. Bringing atom probe tomography to transmission electron microscopes. *Nature communications*, 15:9870, 2024.

12. Ralph Howard Fowler. *Statistical thermodynamics*. CUP Archive, 1939.

13. AM Garrett and CP Race. Cu nanoprecipitate morphologies and interfacial energy densities in bcc Fe from density functional theory (dft). *Computational Materials Science*, 188:110149, 2021.

14. Baptiste Gault, D Haley, Frederic De Geuser, Michael P Moody, Emmanuelle A Marquis, David J Larson, and Brian P Geiser. Advances in the reconstruction of atom probe tomography data. *Ultramicroscopy*, 111(6):448-457, 2011.

15. H Grimmer, WT Bollmann, and DH Warrington. Coincidence-site lattices and complete pattern-shift in cubic crystals. *Acta Crystallographica Section A: Crystal Physics, Diffraction, Theoretical and General Crystallography*, 30(2):197-207, 1974.
16. Raheleh Hadian, Blazej Grabowski, and Jörg Neugebauer. Gb code: A grain boundary generation code. *The Journal of Open Source Software*, 3, 2018.

17. Jian Han, Vaclav Vitek, and David J Srolovitz. Grain-boundary metastability and its statistical properties. *Acta Materialia*, 104:259-273, 2016.

18. Patrick Harrison, Saurabh Mohan Das, William Goncalves, Alessandra da Silva, Xinren Chen, Nicola Viganò, Christian H Liebscher, Wolfgang Ludwig, Xuyang Zhou, and Edgar F Rauch. Determination of five-parameter grain boundary characteristics in nanocrystalline Ni-W by scanning precession electron diffraction tomography. *Ultramicroscopy*, page 114038, 2024.

19. Patrick Harrison, Xuyang Zhou, Saurabh Mohan Das, Pierre Lhuissier, Christian H. Liebscher, Michael Herbig, Wolfgang Ludwig, and Edgar F. Rauch. Reconstructing dual-phase nanometer scale grains within a pearlitic steel tip in 3D through 4D-scanning precession electron diffraction tomography and automated crystal orientation mapping. *Ultramicroscopy*, 238:113536, 2022.

20. Qiongyao He, Søren Schmidt, Wanquan Zhu, Guilin Wu, Tianlin Huang, Ling Zhang, Dorte Juul Jensen, Zongqiang Feng, Andrew Godfrey, and Xiaoxu Huang. 3d microscopy at the nanoscale reveals unexpected lattice rotations in deformed nickel. *Science*, 382(6674):1065-1069, 2023.

21. Olof C Hellman, John Blatz Du Rivage, and David N Seidman. Efficient sampling for three-dimensional atom probe microscopy data. *Ultramicroscopy*, 95:199-205, 2003.

22. M. Herbig, D. Raabe, Y. J. Li, P. Choi, S. Zaefferer, and S. Goto. Atomic-scale quantification of grain boundary segregation in nanocrystalline material. *Physical review letters*, 112:126103, 2014.

23. JP Hirth and RW Balluffi. On grain boundary dislocations and ledges. *Acta Metallurgica*, 21(7):929-942, 1973.

24. Th Karakostas, G Nouet, and P Delavignette. Grain boundary analysis in tem. ii. a modified o-lattice for csl and dsc lattice determination. *physica status solidi* (a), 52(1):65-76, 1979.

25. Lena Langenohl, Tobias Brink, Gunther Richter, Gerhard Dehm, and Christian H. Liebscher. Atomic-resolution observations of silver segregation in a [111] tilt grain boundary in copper. *Physical Review B*, 107:134112, 2023.

26. Irving Langmuir. The adsorption of gases on plane surfaces of glass, mica and platinum. *Journal of the American Chemical society*, 40(9):1361-1403, 1918.

27. Markus Lazar. Non-singular dislocation continuum theories: strain gradient elasticity vs. peierls-nabarro model. *Philosophical Magazine*, 97(34):3246-3275, 2017.

28. Barnaby DA Levin, Yi Jiang, Elliot Padgett, Shawn Waldon, Cory Quammen, Chris Harris, Utkarsh Ayachit, Marcus Hanwell, Peter Ercius, David A Muller, et al. Tutorial on the visualization of volumetric data using tomviz. *Microscopy Today*, 26(1):12-17, 2018.

29. Christian H Liebscher, Andreas Stoffers, Masud Alam, Liverios Lymperakis, Oana Cojocaru-Mirédin, Baptiste Gault, Jörg Neugebauer, Gerhard Dehm, Christina Scheu, and Dierk Raabe. Strain-induced
asymmetric line segregation at faceted si grain boundaries. *Physical review letters*, 121(1):015702, 2018.

30. Jian Luo, Huikai Cheng, Kaveh Meshinchi Asl, Christopher J Kiely, and Martin P Harmer. The role of a bilayer interfacial phase on liquid metal embrittlement. *Science*, 333(6050):1730-1733, 2011.

31. Donald McLean. *Grain Boundaries in Metals*. Monographs on the physics and chemistry of materials. Clarendon Press, 1957.

32. Thorsten Meiners, JM Duarte, Gunther Richter, Gerhard Dehm, and CH Liebscher. Tantalum and zirconium induced structural transitions at complex [111] tilt grain boundaries in copper. *Acta Materialia*, 190:93-104, 2020.

33. Thorsten Meiners, Timofey Frolov, Robert E Rudd, Gerhard Dehm, and Christian H Liebscher. Observations of grain-boundary phase transformations in an elemental metal. *Nature*, 579(7799):375-378, 2020.

34. MK Miller and MG Hetherington. Local magnification effects in the atom probe. *Surface Science*, 246(1-3):442-449, 1991.

35. Heather A Murdoch and Christopher A Schuh. Estimation of grain boundary segregation enthalpy and its role in stable nanocrystalline alloy design. *Journal of Materials Research*, 28(16):2154-2163, 2013.

36. Lawrence Eugene Murr. *Interfacial Phenomena in Metals and Alloys*. Addison-Wesley Publishing Company, Advanced Book Program, 1975.

37. Lawrence Eugene Murr, Richard J Horylev, and Wen N Lin. Interfacial energy and structure in FCC metals and alloys. *The Philosophical Magazine: A Journal of Theoretical Experimental and Applied Physics*, 22(177):515-542, 1970.

38. Jian Feng Nie, YM Zhu, JZ Liu, and Xi-Ya Fang. Periodic segregation of solute atoms in fully coherent twin boundaries. *Science*, 340(6135):957-960, 2013.

39. Jonathan L Priedeman and Gregory B Thompson. The influence of alloying in stabilizing a faceted grain boundary structure. *Acta Materialia*, 201:329-340, 2020.

40. Caihao Qiu, Maik Punke, Yuan Tian, Ying Han, Siqi Wang, Yishi Su, Marco Salvalaglio, Xiaoqing Pan, David J Srolovitz, and Jian Han. Grain boundaries are brownian ratchets. *Science*, 385(6712):980-985, 2024.

41. Dierk Raabe, Michael Herbig, Stefanie Sandlöbes, Yujiao Li, Darius Tytko, Margarita Kuzmina, Dirk 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(4):253-261, 2014.

42. Armin Rajabzadeh, Frédéric Mompiou, S Lartigue-Korinek, Nicolas Combe, Marc Legros, and DA Molodov. The role of disconnections in deformation-coupled grain boundary migration. *Acta Materialia*, 77:223-235, 2014.
43. Edgar F Rauch, Patrick Harrison, Xuyang Zhou, Michael Herbig, Wolfgang Ludwig, and Muriel Véron. New features in crystal orientation and phase mapping for transmission electron microscopy. *Symmetry*, 13(09):1675, 2021.

44. Edgar F. Rauch, Joaquin Portillo, Stavros Nicolopoulos, Daniel Bultreys, Sergei Rouvimov, and Peter Moeck. Automated nanocrystal orientation and phase mapping in the transmission electron microscope on the basis of precession electron diffraction. *Zeitschrift für Kristallographie*, 225(2-3):103–109, 2010.

45. E.F. Rauch and M. Véron. Analyzing dislocations with virtual dark field images reconstructed from electron diffraction patterns. *Microscopy and Microanalysis*, 20(S3):1456–1457, 2014.

46. MP Seah and ED Hondros. Grain boundary segregation. *Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences*, 335(1601):191–212, 1973.

47. DN Seidman, BW Krakauer, and D Udler. Atomic scale studies of solute-atom segregation at grain boundaries: experiments and simulations. *Journal of Physics and Chemistry of Solids*, 55(10):1035–1057, 1994.

48. AN Stroh. Dislocations and cracks in anisotropic elasticity. *Philosophical magazine*, 3(30):625–646, 1958.

49. X.Y. Sun, C. Fressengeas, V. Taupin, P. Cordier, and N. Combe. Disconnections, dislocations and generalized disclinations in grain boundary ledges. *International Journal of Plasticity*, 104:134–146, 2018.

50. A Tehranchi, S Zhang, A Zendegani, C Scheu, T Hickel, and J Neugebauer. Metastable defect phase diagrams as roadmap to tailor chemically driven defect formation. *Acta Materialia*, 277:120145, 2024.

51. K Thompson, D Lawrence, DJ Larson, JD Olson, TF Kelly, and B Gorman. In situ site-specific specimen preparation for atom probe tomography. *Ultramicroscopy*, 107(2-3):131–139, 2007.

52. Keith Thompson, Philip L Flaitz, Paul Ronsheim, David J Larson, and Thomas F Kelly. Imaging of arsenic Cottrell atmospheres around silicon defects by three-dimensional atom probe tomography. *Science*, 317(5843):1370–1374, 2007.

53. Yuan Tian, Xiaoguo Gong, Mingjie Xu, Caihao Qiu, Ying Han, Yutong Bi, Leonardo Velasco Estrada, Evgeniy Boltynjuk, Horst Hahn, Jian Han, et al. Grain rotation mechanisms in nanocrystalline materials: Multiscale observations in pt thin films. *Science*, 386(6717):49–54, 2024.

54. Thomas CT Ting. *Anisotropic elasticity: theory and applications*, volume 45. Oxford university press, 1996.

55. Jason R Trelewicz and Christopher A Schuh. Grain boundary segregation and thermodynamically stable binary nanocrystalline alloys. *Physical Review B—Condensed Matter and Materials Physics*, 79(9):094112, 2009.
56. Stéfan van der Walt, Johannes L. Schönberger, Juan Nunez-Iglesias, François Boulogne, Joshua D. Warner, Neil Yager, Emmanuelle Gouillart, Tony Yu, and the scikit-image contributors. *scikit-image*: image processing in Python. *PeerJ*, 2:e453, 6 2014.

57. Aurélien Vattré and Ernian Pan. Interaction between semicoherent interfaces and volterra-type dislocations in dissimilar anisotropic materials. *Journal of Materials Research*, 32:3947-3957, 2017.

58. Nicola Roberto Viganò and Vicente Armando Solé. Physically corrected forward operators for induced emission tomography: a simulation study. *Measurement Science and Technology*, 29(3):034005, 2018.

59. Malik Wagih, Peter M Larsen, and Christopher A Schuh. Learning grain boundary segregation energy spectra in polycrystals. *Nature communications*, 11(1):6376, 2020.

60. Malik Wagih and Christopher A. Schuh. Grain boundary segregation beyond the dilute limit: Separating the two contributions of site spectrality and solute interactions. *Acta Materialia*, 199:63-72, 2020.

61. Lihua Wang, Yin Zhang, Zhi Zeng, Hao Zhou, Jian He, Pan Liu, Mingwei Chen, Jian Han, David J Srolovitz, Jiao Teng, et al. Tracking the sliding of grain boundaries at the atomic scale. *Science*, 375(6586):1261-1265, 2022.

62. W Xu, B Zhang, XY Li, and K Lu. Suppressing atomic diffusion with the schwarz crystal structure in supersaturated al-mg alloys. *Science*, 373(6555):683-687, 2021.

63. T Yang, YL Zhao, WP Li, CY Yu, JH Luan, DY Lin, Lei Fan, ZB Jiao, WH Liu, XJ Liu, et al. Ultrahigh-strength and ductile superlattice alloys with nanoscale disordered interfaces. *Science*, 369(6502):427-432, 2020.

64. Lan Yao, SP Ringer, JM Cairney, and Michael K Miller. The anatomy of grain boundaries: Their structure and atomic-level solute distribution. *Scripta Materialia*, 69(8):622-625, 2013.

65. Zhiyang Yu, Patrick R Cantwell, Qin Gao, Denise Yin, Yuanyao Zhang, Naixie Zhou, Gregory S Rohrer, Michael Widom, Jian Luo, and Martin P Harmer. Segregation-induced ordered superstructures at general grain boundaries in a nickel-bismuth alloy. *Science*, 358(6359):97-101, 2017.

66. Xiaopu Zhang, Jian Han, John J Plombon, Adrian P Sutton, David J Srolovitz, and John J Boland. Nanocrystalline copper films are never flat. *Science*, 357(6349):397-400, 2017.

67. Huan Zhao, Liam Huber, Wenjun Lu, Nicolas J Peter, Dayong An, Frédéric De Geuser, Gerhard Dehm, Dirk Ponge, Jörg Neugebauer, Baptiste Gault, et al. Interplay of chemistry and faceting at grain boundaries in a model al alloy. *Physical review letters*, 124(10):106102, 2020.

68. Xuyang Zhou, Ali Ahmadian, Baptiste Gault, Colin Ophus, Christian H Liebscher, Gerhard Dehm, and Dierk Raabe. Atomic motifs govern the decoration of grain boundaries by interstitial solutes. *Nature communications*, 14(1):3535, 2023.
69. Xuyang Zhou, Jaber Rezaei Mianroodi, Alisson Kwiatkowski da Silva, Thomas Koenig, Gregory B Thompson, Pratheek Shanthraj, Dirk Ponge, Baptiste Gault, Bob Svendsen, and Dierk Raabe. The hidden structure dependence of the chemical life of dislocations. *Science Advances*, 7(16):eabf0563, 2021.

70. Xuyang Zhou, Ye Wei, Markus Kühbach, Huan Zhao, Florian Vogel, Reza Darvishi Kamachali, Gregory B. Thompson, Dierk Raabe, and Baptiste Gault. Revealing in-plane grain boundary composition features through machine learning from atom probe tomography data. *Acta Materialia*, 226:117633, 2022.

71. Q. Zhu, S.C. Zhao, C. Deng, X.H. An, K.X. Song, S.X. Mao, and J.W. Wang. In situ atomistic observation of grain boundary migration subjected to defect interaction. *Acta Materialia*, 199:42–52, 2020.