
==== Front
Proc Natl Acad Sci U S A
Proc Natl Acad Sci U S A
PNAS
Proceedings of the National Academy of Sciences of the United States of America
0027-8424
1091-6490
National Academy of Sciences

38478685
202400161
10.1073/pnas.2400161121
videoVideoresearch-articleResearch ArticleengEngineering416
Physical Sciences
Engineering
Grain boundary plasticity initiated by excess volume
Zhu Qi a b 1 https://orcid.org/0000-0002-9215-6445

Zhao Qingkun c 1 https://orcid.org/0000-0002-3510-4825

Huang Qishan c 1 https://orcid.org/0000-0003-1769-9340

Chen Yingbin a
Suresh Subra b d https://orcid.org/0000-0002-6223-6831

Yang Wei c
Zhang Ze zezhang@zju.edu.cn
a 2
Zhou Haofei haofei_zhou@zju.edu.cn
c 2 https://orcid.org/0000-0001-9226-9530

Gao Huajian gao.huajian@tsinghua.edu.cn
b e 2 3 https://orcid.org/0000-0002-8656-846X

Wang Jiangwei jiangwei_wang@zju.edu.cn
a f 2 https://orcid.org/0000-0003-1191-0782

aCenter of Electron Microscopy, State Key Laboratory of Silicon and Advanced Semiconductor Materials, School of Materials Science and Engineering, Zhejiang University, Hangzhou 310027, People’s Republic of China
bSchool of Mechanical and Aerospace Engineering, College of Engineering, Nanyang Technological University, Singapore 639798, Singapore
cDepartment of Engineering Mechanics, Center for X-Mechanics, Zhejiang University, Hangzhou 310027, People’s Republic of China
dDepartment of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139
eInstitute of High Performance Computing, Agency for Science, Technology and Research (A*STAR), Singapore 138632, Singapore
fWenzhou Key Laboratory of Novel Optoelectronic and Nano Materials, Institute of Wenzhou, Zhejiang University, Wenzhou 325006, People’s Republic of China
2To whom correspondence may be addressed. Email: zezhang@zju.edu.cn, haofei_zhou@zju.edu.cn, gao.huajian@tsinghua.edu.cn, or jiangwei_wang@zju.edu.cn.
Contributed by Huajian Gao; received January 5, 2024; accepted February 13, 2024; reviewed by John P. Hirth and Marc A. Meyers

1Q. Zhu, Q. Zhao, and Q.H. contributed equally to this work.

3Present address: Department of Engineering Mechanics, Center for Advanced Mechanics and Materials, Applied Mechanics Laboratory, Tsinghua University, Beijing 100084, People’s Republic of China.

13 3 2024
19 3 2024
13 9 2024
121 12 e240016112105 1 2024
13 2 2024
Copyright © 2024 the Author(s). Published by PNAS.
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

Significance

In crystalline solids, the presence of grain boundaries (GBs), the interfaces between individual grains, introduces inherent excess volume due to loose atomic packing. This excess volume makes GBs more susceptible to external stimuli, including a proneness to accommodating plastic deformation under mechanical loading. While some aspects of GB deformation, such as Navarro–Herring creep and shear-driven GB migration, have already been treated, the dilatational effect of GB-accommodated deformation associated with GB excess volume has long remained elusive due to its transient nature. Here, by using state-of-the-art experimental and computational methods, we unveil a general route by which GB plasticity can be initiated by its excess volume. This work provides a key missing piece for understanding GB-associated plastic deformation mechanisms.

Grain boundaries (GBs) serve not only as strong barriers to dislocation motion, but also as important carriers to accommodate plastic deformation in crystalline solids. During deformation, the inherent excess volume associated with loose atomic packing in GBs brings about a microscopic degree of freedom that can initiate GB plasticity, which is beyond the classic geometric description of GBs. However, identification of this atomistic process has long remained elusive due to its transient nature. Here, we use Au polycrystals to unveil a general and inherent route to initiating GB plasticity via a transient topological transition process triggered by the excess volume. This route underscores the general impact of a microscopic degree of freedom which is governed by a stress-triaxiality-based criterion. Our findings provide a missing perspective for developing a more comprehensive understanding of the role of GBs in plastic deformation.

grain boundary
excess volume
plasticity
stress triaxiality
MOST | National Key Research and Development Program of China (NKPs) 501100012166 2021YFA1200201 Haofei ZhouJiangwei Wang MOST | National Key Research and Development Program of China (NKPs) 501100012166 2022YFB3707401 Haofei ZhouJiangwei Wang MOST | National Natural Science Foundation of China (NSFC) 501100001809 52071284 51771172 12222210 12172324 Haofei ZhouJiangwei Wang Agency for Science, Technology and Research (A*STAR) 501100001348 M22L2b0111 Qi ZhuHuajian Gao
==== Body
pmcDeformation of crystalline solids can be accommodated either uniformly by the crystal lattices (Fig. 1A) or preferentially at the crystal defects (1). As common defects sustaining the structural integrity and mass transport in polycrystalline materials (2, 3), grain boundaries (GBs) have been widely recognized as not only strong barriers to dislocation motion, but also carriers to accommodate plastic deformation (4–7), thereby modulating the mechanical behaviors of materials (8). In classic theories, the GB plasticity in response to external loading is mediated by the sequential nucleation, propagation, and interaction of GB defects (e.g., disconnections), leading to GB migration or GB sliding (9–12). GBs can also serve as preferential nucleation sites of lattice defects (e.g., dislocations or twins) that propagate into the grains (13, 14), which underscores GB sources as initiators for plasticity (15). These different modes of GB plasticity are generally acknowledged to be governed by the five macroscopic geometric degrees of freedom of GBs (16).

Fig. 1. GB plastic deformation initiated by excess volume through TTT. (A) Uniform lattice straining in response to external load in crystalline solid. (B) Plastic deformation initiated preferentially at the GB through excess-volume-assisted TTT. The red or blue circles denote atoms that possess an excess volume above or below that of the perfect lattice (black circles). (C–E) In situ high-resolution TEM snapshots showing TTT at a general high-angle GB in Au with increasing compressive loading. (C) Atomic structure of a 24° [11¯0] asymmetrical tilt GB in Au. Two alternate types of facets are delineated by the white and yellow dashed lines. (D) TTT of the originally faceted GB into an orthogonal configuration (highlighted yellow) under compressive loading (indicated by the black arrow). The TTT mainly occurs at the upper edge of GB. (E) Continuous expansion of the TTT region along the GB with limited thickening into G2. (F–I) TTT at a curved GB in polycrystalline Au. (F and G) Atomic structures of the flat and curved segments of the initial 32° [11¯0] tilt GB. (H and I) TTT at the flat and curved GB segments under [112]G1-compression (indicated by the black arrow). (Scale bar: 2 nm.)

Beyond the widely known GB geometry, the inherent excess volume of GBs, as compared to a close-packed lattice (17, 18), provides extra space that can stimulate structural transitions along the GBs prior to the onset of conventional plastic deformation modes under external loading (schematically illustrated in Fig. 1B). Whether this dynamic process governed by the excess volume, a microscopic degree of freedom, can bring about an alternative route to initiating GB plasticity is an issue that has eluded detailed experimental evidence and analysis. In principle, the GB configurations under external stimuli (e.g., mechanical loading, temperature gradient, or electric field) can frequently deviate from their equilibrium counterparts and assume metastable structures (19–21) that influence GB responses and materials properties. Recent studies have shown different atomic-scale GB variants in face-centered cubic (FCC) and body-centered cubic (BCC) metals (22, 23). However, the general impact of inherent excess volume on the local structural transition of GBs and subsequent GB plasticity remains unclear. Such inherent GB structural transition can be highly localized and transient, which can relax instantly with changing stress state or merge with conventional deformation mechanisms such as GB migration and dislocation nucleation. These challenges often make it difficult to clarify, by recourse to experiment, different dynamic responses of GBs and their critical roles in modulating plasticity.

Here we unveil the atomistic dynamics of a GB plastic deformation mechanism initiated by excess volume at room temperature through a transient topological transition (TTT) process, which precedes conventional GB deformation modes, using in situ atomic-resolution transmission electron microscopy (TEM) during nanomechanical testing, and atomistic simulations assisted by machine learning. A stress-triaxiality-based criterion is developed to elucidate the impact of this extra degree of freedom, which provides a key missing perspective for developing the full map of GB plasticity when coordinated with conventional GB deformation mechanisms.

Results

Fig. 1 C–E presents a typical example to demonstrate excess-volume-assisted GB structural transition and plastic deformation. Fig. 1C shows a 24° [11¯0] asymmetrical tilt GB in an Au bicrystal (Materials and Methods and SI Appendix, Fig. S1), which consists of alternate facets of horizontal type (1¯1¯1)G1||(227¯)G2 and inclined type (331¯)G1||(1¯1¯1)G2. Upon compressive loading applied on the bottom grain along the [1¯1¯1]G1 direction, an evident GB structural transition (i.e., TTT) is induced preferentially in this faceted GB at the onset of plastic deformation (Fig. 1D and Movie S1), with negligible trace of commonly observed deformation modes such as GB migration or dislocation emission (6, 13, 24). Specifically, TTT in the middle segment of this tilt GB exhibits a typical orthogonal structure parallel to the respective (002)G1 and (1¯1¯1)G2 lattice planes of the neighboring grains. Atomistic analysis indicates that the characteristic interplanar spacings inside this TTT region (see the Inset in Fig. 1D) differ from those of any projected lattice planes of FCC-Au and the lattice overlap across GB (see the high-resolution TEM simulations in SI Appendix, Fig.S2). In contrast, the lattices inside grains exhibit negligible variation, which largely remain in the elastic regime. Such inhomogeneous structural change indicates that plastic deformation is preferentially initiated by TTT at the GB. With continuous compression, this TTT region expands both laterally along the GB and inward along the upper grain G2 (Fig. 1E), leading to an upward migration of this GB. Meanwhile, the lattice inside the TTT region tends to evolve into a tetragonal structure associated with an apparent extension along the [112]G2 direction by 4.7% and a contraction along the [1¯1¯1]G2 direction by 3.2% (Fig. 1E). Machine-learning-assisted atomic stress maps converted from atomic-resolution TEM snapshots in Fig. 1 C and E confirm that the TTT-initiated plasticity is governed by preferential stress accumulation at the GB (Materials and Methods and SI Appendix, Fig. S3 A and B), which clarifies the absence of intragranular dislocation activity throughout this process, despite the availability of favorable slip orientations. Apart from the fully developed TTT in the middle, the transitional lattice near the edge of the TTT region deviates from the orthogonal structure. This non-uniform TTT arises from the local variation of stress concentration along the GB (SI Appendix, Fig. S3 A and B). SI Appendix, Fig. S3 C–E further demonstrates the atomic structure of the evolving lattices ahead of the GB segment modified by TTT, which can serve as an activation site for subsequent plastic deformation via GB migration. When the loading is changed to horizontal shear, the GB TTT recovers instantly, leaving an atomically sharp 24° [11¯0] GB (SI Appendix, Fig. S1F). In view of the dependence of this recoverable GB structural evolution on external loading condition, such GB TTT, despite its possibly not-so-uncommon occurrence in GBs, could have been easily missed in previous studies, where in situ atomic-resolution electron microscopy could not be used in real-time plastic deformation.

With the excess volume as an inherent microscopic degree of freedom of GBs, such TTT-initiated plasticity should have a general presence among different GBs. Fig. 1 F–I shows another example in 32° [11¯0] tilt GB with continuously changing curvature in polycrystalline Au (see SI Appendix, Fig. S4 A and B for the overall GB morphology), where two representative segments of this GB, i.e., flat and curved, are traced in real time during the compressive loading. Prior to loading, both flat and curved segments are atomically sharp (Fig. 1 F and G). During compression along [112]G1, TTT is evidently activated at the onset of plastic deformation, which expands laterally along the GB and grows preferentially into grain G1 (Fig. 1H and SI Appendix, Fig. S4C). Similar TTT occurs at the curved GB segment as well (Fig. 1I), where the atomic structures of the TTT region and its crystallographic orientations with respect to the neighboring grains are identical to those at the flat segments. Other potential GB deformation modes, such as dislocation nucleation or GB sliding, have not been observed during the compression. When compressive loading is gradually released, the TTT recovered, leaving behind an atomically sharp GB; with reversed loading, GB migration and partial dislocation nucleation from the GB occurs (SI Appendix, Fig. S4D and Movie S2). This demonstrates once more the transient nature of this GB TTT and confirms the indispensable role of compressive stress in reducing GB excess volume. Moreover, the crystallographic relations between the TTT-modified GB and the (1¯1¯1)G1 and (002)G2 lattice planes in the neighboring grains show certain similarities with those of the 24° [11¯0] GB shown in Fig. 1E, which depict a potentially general route to initiate plasticity through TTT among different GBs.

To quantitatively clarify the underlying dynamics of the excess-volume-initiated GB plasticity, molecular dynamics (MD) simulations were performed to reveal the atomistic mechanism from both energetic and kinetic perspectives (Materials and Methods). Fig. 2A demonstrates the structural evolution of 24° [11¯0] tilt GB in Au under compressive loading, matching well with our experimental observations (Fig. 1 D and E). During this GB plasticity initiation process associated with TTT, an evident shear displacement of the close-packed {111} lattice planes in G2 is activated with the compressive strain, as demonstrated by the superimposed atomistic snapshots (Fig. 2B). The overall crystallographic relations among the GB TTT and neighboring grains in the two examples shown in Fig. 1 are further illustrated by the trichromatic patterns (SI Appendix, Fig. S5), clarifying the preference of GB TTT along one of the neighboring grains. Such structural evolution of GB is governed by the instantaneous reduction of GB excess volume at the onset of compressive loading (SI Appendix, Fig. S6 A and B), which leads to the TTT-initiated GB plasticity under loading conditions comparable to those of dislocation nucleation at room temperature, in contrast to the FCC-to-BCC phase transformation in Au single crystals induced by laser-shock over 150 GPa with concomitant high temperatures (25). Moreover, the TTT-mediated GB plasticity observed in Au (Fig. 1 C–E) also occurs in FCC Ni and Al containing the same 24° [11¯0] tilt GB (see the MD simulation results in SI Appendix, Fig. S6 D and E), although different extents of TTT form due to distinctly different excess volumes among these metals (18). Nevertheless, the eventual crystallographic features of GB after TTT are essentially affected by the distinct orientations of low-index lattice planes such as {002} and {111} in the neighboring grains. They can assume a wide range of configurations beyond the cases demonstrated here. By comparison, the GB plane orientation appears to exert less influence on TTT, as indicated by similar TTT structures along a curved GB with varying inclinations (Fig. 1 H and I).

Fig. 2. Atomistic origin of excess volume governed initiation of GB plasticity. (A) Compression-induced TTT at the 24° [11¯0] tilt GB in Au in MD simulation. The black and red profiles delineate the original GB and the advancing front of TTT, respectively. (B) Tracked displacements of atom columns at the GB with increasing compressive strain. (C) Emission of a partial dislocation (bound by a stacking fault, SF) following TTT at the 24° [11¯0] tilt GB in Au under compression. (D) High-resolution TEM snapshots showing similar partial dislocation emission from the 32° [11¯0] tilt GB in Au after TTT under compression. Scale bars: (A) 1 nm; (C and D) 2 nm. (E) Statistical mapping of incipient GB plastic deformation modes governed by the local stress state at GBs, which is dependent on excess volume. This stress can be decomposed into hydrostatic pressure σhydrostatic and a deviatoric stress component that is usually evaluated by the von-Mises stress σvon-Mises. The stress triaxiality factor, σhydrostatic/σvon-Mises, is adopted to quantitatively characterize stress states that initiate GB plastic deformation (error bars represent the SDs from statistical analyses, where n = 5 for averaged value among five sequential data points for the onset of GB TTT or dislocation emission).

Besides the atomic-scale structural transition at preferential GB regions, the development of GB TTT can also stimulate further GB-associated plastic deformation that we commonly observe, such as dislocation nucleation from GB. As indicated by MD simulations (Fig. 2C), a flat 24° [11¯0] tilt GB first underwent TTT associated with a local reduction of GB excess volume from 0.13 Å to −0.01 Å (conventionally defined as the extra volume of GB divided by the cross-sectional area of GB), and the local structure modification at GB facilitates the nucleation of a partial dislocation during subsequent compressive loading. Similar TTT-initiated dislocation nucleation is also observed in our experiment (see the 32° [11¯0] tilt GB in Fig. 2D). Here, TTT at the horizontal GB segment continues to expand while the TTT at inclined GB segment stimulates the emission of a partial dislocation into the upper grain, enabling the conventional mode of plastic deformation.

In contrast to the classic GB theory based on macroscopic geometric degrees of freedom, the mechanism of TTT-initiated GB plastic deformation presented here highlights the important role of microscopic degrees of freedom at GBs. They endow distinct extra spaces for the evolution of GB structure that can compete or coordinate with conventional plastic deformation modes. To further elucidate the general significance of this microscopic degree of freedom from the kinetic perspective, statistical analysis of incipient plastic deformation at different GBs in polycrystalline Au was carried out in MD simulations (SI Appendix, Fig. S7). These simulations indicate that a general correlation between large GB excess volume and TTT-initiated plasticity can be established (SI Appendix, Table S1). This intrinsic dependence of GB TTT on the inherent GB excess volume is further clarified by the statistical map in Fig. 2E, where the local stress state of GB arising from external loading serves as the key bridge (SI Appendix). Theoretically, the local stress state of GBs can be quantified by the stress triaxiality factor σhydrostatic/σvon-Mises, which characterizes the competition between changes in volume and shape during plastic deformation. Statistical analyses among different GBs demonstrate a direct correlation between the stress triaxiality factor and the initial excess volume of GBs. This result underscores GB TTT as a necessary volumetric evolution process and rationalizes its preference at GBs with larger excess volume. In contrast, dislocation emission from the GB depends on the local resolved shear stress and thus appears to be more sensitive to the von-Mises stress, which is favored at GBs with lower excess volume (e.g., Shockley partial emitted from the twin boundary with negligible excess volume).

Accumulating evidence further indicates the general significance of the excess-volume-initiated TTT to GB plasticity in polycrystalline metals. Fig. 3 A–C show an example of TTT process near the triple junction in a GB network in a pre-deformed polycrystalline Au. Before loading, several TTT clusters are found at the steps in a 90° [11¯0] general high-angle GB (denoted as GB2-3, Fig. 3A), indicating the general presence of GB TTT in pre-deformed polycrystals. Additional examples of residual TTT configurations at GBs in deformed bulk polycrystalline Au can be found in SI Appendix, Fig. S8, where the atomic structures of both continuous and localized TTT associated with reduced GB excess volume are determined by high-angle annular dark field scanning transmission electron microscopy (HAADF-STEM, see Materials and Methods). Upon further loading, these TTT clusters interlink with one another, followed by continuous expansion along GB2-3 (Fig. 3 B and C), initiating evident plastic deformation of the GB network. Similar TTT nucleation and expansion processes, and their stimulation of subsequent GB migration (a common mode of GB plasticity in different materials) are confirmed in different GB networks (SI Appendix, Fig. S9).

Fig. 3. Generalized GB TTT and its coupling with conventional GB defects. (A–C) TTT near a GB triple junction in pre-deformed polycrystalline Au. (A) TTT clusters pre-exist at the steps in a 90° [11¯0] high-angle GB2-3 (marked with aqua circles). (B and C) Interlink and continuous expansion of TTT in GB2-3 under shear loading. Inset FFT pattern confirms the orthogonal-structured TTT at GB2-3. (D–F) TTT initiates dislocation-mediated GB migration in Au. (D and E) Dynamic transformation from orthogonal TTT to well-aligned dislocation array in a 21° [11¯0] tilt GB under tensile loading. (F) Subsequent GB migration via the collective motion of dislocations. (G–I) GB TTT associated with the migration of a [112](22¯0)//[11¯0](113) mixed GB in Au under compression. (G) TTT coupled with a pre-existed disconnection (D1). (H and I) Continuous shrinking of TTT that adheres to D1 during downward migration of the right GB segment (shown by the white arrow). Stress evolution associated with TTT shrinkage facilitates the dynamic composition and decomposition of disconnections D2 and D3 in the left GB segment. (J) Atomic structure of TTT at the core of D1. Scale bars: (A–C and G–I) 2 nm; (D–F and J) 1 nm.

In addition to direct accommodation of incipient plastic deformation, this dynamic process of GB TTT can also coordinate with other classic modes of GB deformation and catalyze subsequent GB plasticity. Fig. 3 D–F show an example of the transformation of GB TTT into a dislocation array via stress-induced local lattice relaxation, which facilitates subsequent GB migration via collective dislocation motion. Since dislocation dynamics also depends on GB excess volume (26), GB TTT can also facilitate the emission of dislocations from the GB (Fig. 2 C and D and SI Appendix, Fig. S7). In high-angle GBs, prevalent GB defects (e.g., disconnections with distinct Burgers vector and step characters) readily build up stress concentration and can change the local GB excess volume, which are likely to favor the local TTT alongside GB migration. Fig. 3 G–J exemplifies a TTT process that is coordinated with conventional disconnection kinetics upon compression-induced migration of a [112](22¯0)//[11¯0](113) mixed GB in Au. Specifically, the core of a pre-existing multi-atom-layer disconnection in the horizontal GB exhibits a parallelogram configuration (Fig. 3G). Here, the characteristic lattices inside the TTT region align with the (22¯0)G1 and (111¯)G2 lattice planes in the respective grains (Fig. 3J), similar to those at tilt GBs shown in Fig. 1. Under compressive loading, this TTT-coupled sessile disconnection core shrinks continuously due to the annihilation of smaller disconnections that mediate the migration of right GB segment (Fig. 3 H and I). Meanwhile, the shrinking of TTT region in turn facilitates the dynamic interaction of nearby disconnections (D2 and D3) on the left GB segment, smoothing the continuous migration of the entire mixed GB (Fig. 3I). Therefore, the coupled GB TTT and conventional GB defect dynamics is of general significance to GB plasticity.

Discussion and Conclusion

From a general perspective, the atomic structures of GBs under external loading commonly deviate from their equilibrium counterparts predicted by the classic theory of five macroscopic geometric degrees of freedom. Due to different extents of stress-excess volume coupling, multiple types of GB configurations, either well-developed TTT (Fig. 1) or distorted structural units (23, 27, 28), can co-exist on the GBs, readily accommodating the incipient GB plasticity and potentially modulating subsequent deformability of GBs (29). On the other hand, the TTT-modified GB configurations also facilitate the nucleation of GB defects, which can influence subsequent plastic deformation. For example, the junctions of different GB phases in elemental metals can be regarded as line defects with distinct dislocation contents that can influence the thermal and mechanical properties (30, 31). With its effect on the local stress state, GB TTT can catalyze commonly observed plastic deformation modes (32), such as GB migration and dislocation emission from the GB (Figs. 2 and 3 and SI Appendix, Fig. S7).

In summary, GB plasticity is shown here to be initiated by inherent excess volume, which is governed by stress accommodation via GB TTT. This route to GB plasticity is elucidated by a stress-triaxiality-based criterion, and the associated impact on subsequent conventional GB deformation modes also contributes to the GB plasticity. Results of atomic-level dynamics of TTT reported in this work offer unique perspectives on the role of GBs in plastic deformation beyond classic theories, providing a more comprehensive picture of GB plasticity in polycrystalline solids.

Materials and Methods

Au Bicrystal/Polycrystal Sample Preparation.

Au was chosen as an ideal model for in situ TEM nanomechanical testing, taking advantage of its chemical inertness (to avoid the surface oxidation), high atomic weight (to reduce the irradiation damage), and high thermal conductivity (to suppress evident temperature rise). High-purity Au rods obtained from Alfa Aesar Inc. (99.99 wt.%, with a diameter of 0.25 mm) were compressed to a thickness of 0.1 mm and then cut by a sharp wire cutter to obtain a clean fracture surface, which consists of randomly oriented ultra-fine or nanosized grains with different GBs. The electron-transparent samples near the fracture surface possess a thickness of approximately 50 nm, and the ideally edge-on GBs with either tilt or mixed characters in these samples were selected for atomistic characterizations and in situ TEM nanomechanical testing.

In Situ TEM Nanomechanical Testing.

In situ nanomechanical testing was carried out using a PicoFemto® electrical holder from Zeptools Co. inside FEI Titan G2 60 to 300 aberration-corrected TEM operated at 300 kV with a controlled low beam dose rate of approximately 1 × 104 e− nm−2 s−1. Before experiments, the Au bicrystal/polycrystal containing a fracture surface and the Au indenter were placed on the static and mobile sides of the TEM holder, respectively. During in situ experiment, the indenter was first moved via a piezo-controlled actuator to approach the ideal area containing an edge-on GB. Then, the probe was moved carefully to contact with a specific grain exposed to the surface. To ensure a uniaxial loading, the Z position of the probe was adjusted to coincide with that of the sample side before contact. Compressive loading at room temperature was then carried out by precisely controlling the forward displacement of the indenter (toward the sample) at a nominal strain rate of ~10−3 s−1, which can induce GB TTT in the electron-transparent sample. For cyclic loading, the compression was stopped immediately when the GB TTT ceased to expand, followed by tensile loading via the reversed displacement of the indenter. All the in situ experiments were recorded under atomic resolution by a Gatan 994 charge-coupled device camera at a rate of ~0.3 s per frame.

High-Resolution TEM Image Simulations.

High-resolution TEM image simulations were carried out using QSTEM software. The original TEM image was used as the input file, which enabled the construction of an atomic model of ideal Au bicrystal containing a faceted tilt GB (identical to that shown in Fig. 1C). To further construct an overlapped lattice of the neighboring grains at the GB, the lattice of the bottom grain was extended for approximately 2 nm into the top grain near the GB facet. The thickness of the bicrystal cells was set as 20 nm, and a slice thickness of 1 nm was adopted. The acceleration voltage was set to 300 kV with a spherical aberration of 0.012 mm, based on the parameters in experiments. A range of defocus values from +45 nm to −80 nm were used in simulations to export different phase-contrast images for direct comparison with the high-resolution TEM images obtained during in situ experiments.

HAADF-STEM Determination of GB TTT Structures in Bulk Polycrystalline Au.

TEM lamellae were lifted out from the compressed polycrystalline Au rod (with an apparent strain of approximately 60%) along the loading direction using focused ion beam (FIB, ZEISS Crossbeam 540) followed by consecutive thinning using a 30 kV Ga+ ion beam with low currents of 100 pA and 20 pA to reduce irradiation damage. A low voltage polishing (5 kV, 10 pA) was conducted to eliminate potential amorphous layers on both sides, and the eventual thickness of the TEM lamellae was reduced to approximately 30 to 50 nm for atomic-resolution characterizations. Atomic structures of residual TTT at different GBs in polycrystalline Au lamellae were determined using JEOL JEM-ARM300F double aberration-corrected S/TEM operated at 300 kV. HAADF-STEM images were captured with a semi-convergence angle of approximately 28 mrad and a dwell time of 8 s. Consistent residual GB TTT structures were observed in both FIB-fabricated polycrystalline Au samples and near the fracture surface of Au rod without FIB fabrication, which confirm the intrinsic nature of TTT in GB plasticity.

Atomistic Mechanism of GB TTT via MD Simulations.

MD simulations were carried out on Au bicrystal and Au polycrystal models using Large-scale Atomic/Molecular Massively Parallel Simulator (33). The interatomic interactions between Au atoms were modeled with the embedded atom method interatomic potentials (34), and the integration time step was fixed at 2 fs. A bicrystal slab with dimensions of 200 × 30 × 14.6 nm3 was created by joining two perfect nanoscale crystals (with the same orientations as the neighboring grains observed in experiments) along the axial direction. GBs with different misorientations were generated by tilting the top grain of the bicrystal around the <110> tilt axis while fixing the bottom grain. Free boundary condition was applied along the axial direction of the bicrystal, where three layers of atoms at the top and bottom grains were fixed as rigid slabs; meanwhile, the other two orthogonal directions were set periodic. Before compressive loading, the initial bicrystal system was optimized with conjugate gradient minimization method to obtain the equilibrium GB structure. During the compressive loading, a constant velocity of 0.2 m s−1 along the axial direction of the bicrystal was applied on the rigid slab in the top grain under a canonical ensemble with a Nose–Hoover thermostat at 300 K. Similar simulation procedures were also applied in Ni and Al bicrystals (35) containing the same GB as that in Au. The corresponding stress–strain curve was generated by extracting a series of data points at different intervals during compressive loading. Compression of polycrystalline Au was carried out on 40 × 40 × 40 nm3 three-dimensional models containing grains with random misorientations and grain sizes varying from 7 nm to 22 nm, which is indicative of the generality of TTT in various GBs in bulk polycrystalline samples. Ovito (36) was used to visualize the models.

The atomistic volume was calculated as the Voronoi cell volume of each atom, and the atomic excess volume of the i-th atom was obtained with Vexcess-i=Vi-V¯ (where Vi is the Voronoi volume of the i-th atom, and V¯ is the mean Voronoi volume in an FCC single crystal model). The excess volume of GB (eGB) can be expressed as (37, 38): eGB=(V-NΩ)/A, where V is the volume of the system containing a relaxed GB, N is the total number of atoms in the system, Ω is the mean atomistic volume calculated from a single crystal Au model with the same simulation settings as the bicrystal model, and A is the GB area.

To clearly show the stress distribution, the compressive loading was performed under an isothermal isobaric ensemble at a temperature of 10 K maintained with a Nose–Hoover thermostat. The pressure of the two directions orthogonal to the compressive loading was set to zero. The local stress conditions of a grain boundary, σhydrostatic and σvon-Mises, are the mean value of atomic stresses over 4 to 8 layers of atoms across the GB. The discrete atomistic stress field was directly obtained using the Virial theorem (39, 40). The von Mises stress is defined asσvon-Mises=1/2[(σx-σy)2+(σx-σz)2+(σz-σy)2+6(τxy2+τyz2+τxz2)],

where σx, σy, σz, τxy, τyz, and τxz are the six independent components of the stress tensor per-atom that can be obtained directly from MD simulations.

Machine-Learning Assisted Prediction of Atomic Stress.

A machine-learning-based framework Atom-S2 (41) was employed to predict the atomic stress of GBs directly from high-resolution TEM images. First, the atomic information (atom position, compressive stress σzz and hydrostatic pressure) was obtained from MD simulations of compressed 24° [11¯0] tilt GBs. A dataset containing 102,458 data points was established by collecting the atomic information in MD simulation for 5.25 ps (until an engineering strain of −7.5%) at a step interval of 0.25 ps. Smooth Overlap of Atomic Positions (SOAP) descriptor (42, 43) was then used to describe local atomic environment of each atom, where the cut-off distance was set as 6 Å, the number of radial basis functions was set to 3, and the maximum degree of spherical harmonics was 6. As a result, the local atomic environment of each atom was described with 24 descriptors, generating a 102,458 × 24 feature matrix. Third, a backpropagation artificial neural network with three hidden layers (topology 15-15-10) was trained with the Levenberg–Marquardt optimization method (taking 70% of the dataset as training set, 15% as validation set, and 15% as testing set) to predict the atomic stress (σzz and von Mises stress) state from SOAP feature matrix. Finally, the TEM image pre-processing and atomic coordinate recognition was realized by AtomSegNet (44) and the SOAP matrix can be obtained accordingly. Combining the trained network and the high-resolution TEM image-based SOAP matrix, the atomic stress state can be eventually predicted by Atom-S2.

Supplementary Material

Appendix 01 (PDF)

Movie S1. In situ atomistic observation of TTT-initiated plasticity at a 24° [11-0] tilt GB in Au under compressive loading.

Movie S2. In situ atomistic observation of TTT-initiated plasticity at a 32° [11-0] tilt GB with changing curvature in Au under compressive loading.

J.W. and H.Z. acknowledge the support of National Key R&D Program of China (2021YFA1200201 and 2022YFB3707401). J.W. and H.Z. acknowledge the financial support from the National Natural Science Foundation of China (52071284, 51771172, 12222210, and 12172324) and computational support from the Super Cloud Computing Center in Beijing. J.W. acknowledges the Zhejiang Provincial Natural Science Foundation of China (LR24E010002). Q. Zhu and H.G. acknowledge support by A*STAR under its Advanced Models for Additive Manufacturing (AM2) program (Award M22L2b0111). Q. Zhu, S.S., and H.G. acknowledge Nanyang Technological University for partial support of this work through the Distinguished University Professorships for S.S. and H.G. Q. Zhu would like to acknowledge the Facility for Analysis, Characterization, Testing and Simulation, Nanyang Technological University, Singapore, for use of their FIB and TEM.

Author contributions

H.G. and J.W. designed research; Q. Zhu, Q. Zhao, Q.H., Y.C., H.Z., and J.W. performed research; Q. Zhu, Q. Zhao, S.S., W.Y., Z.Z., H.Z., H.G., and J.W. analyzed data; S.S., W.Y., and Z.Z. contributed to paper revision; and Q. Zhu, Q. Zhao, H.Z., H.G., and J.W. wrote the paper.

Competing interests

The authors declare no competing interest.

Data, Materials, and Software Availability

All study data are included in the article and/or supporting information.

Supporting Information

Reviewers: J.P.H., Washington State University; and M.A.M., University of California San Diego.
==== Refs
1 J. Hirth, J. Lothe, Theory of Dislocations (John Wiley & Sons, New York, 1982).
2 K. Lu, Stabilizing nanostructures in metals using grain and twin boundary architectures. Nat. Rev. Mater. 1 , 1–13 (2016).
3 Z. Yu , Segregation-induced ordered superstructures at general grain boundaries in a nickel-bismuth alloy. Science 358 , 97–101 (2017).28983049
4 K. Lu, L. Lu, S. Suresh, Strengthening materials by engineering coherent internal boundaries at the nanoscale. Science 324 , 349–352 (2009).19372422
5 Z. Shan , Grain boundary-mediated plasticity in nanocrystalline nickel. Science 305 , 654–657 (2004).15286368
6 T. J. Rupert, D. S. Gianola, Y. Gan, K. J. Hemker, Experimental observations of stress-driven grain boundary migration. Science 326 , 1686–1690 (2009).20019286
7 P. Cordier , Disclinations provide the missing mechanism for deforming olivine-rich rocks in the mantle. Nature 507 , 51–56 (2014).24572356
8 M. A. Meyers, K. K. Chawla, Mechanical Behavior of Materials (Cambridge University Press, Cambridge, ed. 2, 2008).
9 J. P. Hirth, R. C. Pond, Steps, dislocations and disconnections as interface defects relating to structure and phase transformations. Acta Mater. 44 , 4749–4763 (1996).
10 J. Han, S. L. Thomas, D. J. Srolovitz, Grain-boundary kinetics: A unified approach. Prog. Mater. Sci. 98 , 386–476 (2018).
11 Q. Zhu , In situ atomistic observation of disconnection-mediated grain boundary migration. Nat. Commun. 10 , 156 (2019).30635566
12 L. Wang , Tracking the sliding of grain boundaries at the atomic scale. Science 375 , 1261–1265 (2022).35298254
13 V. Yamakov, D. Wolf, S. R. Phillpot, A. K. Mukherjee, H. Gleiter, Dislocation processes in the deformation of nanocrystalline aluminium by molecular-dynamics simulation. Nat. Mater. 1 , 45–48 (2002).12618848
14 L. E. Murr, Dislocation ledge sources: Dispelling the myth of Frank-Read source importance. Metall. Mater. Trans. A 47 , 5811–5826 (2016).
15 J. C. Li, Petch relation and grain boundary sources. Trans. Metall. Soc. AIME 227 , 239 (1963).
16 A. P. Sutton, R. W. Balluffi, Interfaces in Crystalline Materials (Oxford University Press, Oxford, 1996).
17 E. M. Steyskal , Direct experimental determination of grain boundary excess volume in metals. Phys. Rev. Lett. 108 , 055504 (2012).22400941
18 J. J. Bean, K. P. McKenna, Origin of differences in the excess volume of copper and nickel grain boundaries. Acta Mater. 110 , 246–257 (2016).
19 J. Han, V. Vitek, D. J. Srolovitz, Grain-boundary metastability and its statistical properties. Acta Mater. 104 , 259–273 (2016).
20 J. Nie, C. Hu, Q. Yan, J. Luo, Discovery of electrochemically induced grain boundary transitions. Nat. Commun. 12 , 2374 (2021).33888715
21 J. Wei , Direct imaging of atomistic grain boundary migration. Nat. Mater. 20 , 951–955 (2021).33432148
22 Q. Zhu, A. Samanta, B. Li, R. E. Rudd, T. Frolov, Predicting phase behavior of grain boundaries with evolutionary search and machine learning. Nat. Commun. 9 , 467 (2018).29391453
23 T. Meiners, T. Frolov, R. E. Rudd, G. Dehm, C. H. Liebscher, Observations of grain-boundary phase transformations in an elemental metal. Nature 579 , 375–378 (2020).32188953
24 A. M. Minor , A new view of the onset of plasticity during the nanoindentation of aluminium. Nat. Mater. 5 , 697–702 (2006).16906139
25 S. M. Sharma , Structural transformation and melting in gold shock compressed to 355 GPa. Phys. Rev. Lett. 123 , 045702 (2019).31491271
26 M. A. Tschopp, G. J. Tucker, D. L. McDowell, Structure and free volume of <110> symmetric tilt grain boundaries with the E structural unit. Acta Mater. 55 , 3959–3969 (2007).
27 T. Frolov, D. L. Olmsted, M. Asta, Y. Mishin, Structural phase transformations in metallic grain boundaries. Nat. Commun. 4 , 1899 (2013).23695693
28 J. Han, V. Vitek, D. J. Srolovitz, The grain-boundary structural unit model redux. Acta Mater. 133 , 186–199 (2017).
29 T. Shimokawa, T. Niiyama, T. Miyaki, M. Ikeda, K. Higashida, A novel work hardening mechanism of nanoscale materials by grain boundary transformation. Acta Mater. 224 , 117536 (2021).
30 T. Frolov, D. L. Medlin, M. Asta, Dislocation content of grain boundary phase junctions and its relation to grain boundary excess properties. Phys. Rev. B 103 , 184108 (2021).
31 K. Chen, D. J. Srolovitz, J. Han, Grain-boundary topological phase transitions. Proc. Natl. Acad. Sci. U.S.A. 117 , 33077–33083 (2020).33318180
32 M. A. Meyers, A. Mishra, D. J. Benson, Mechanical properties of nanocrystalline materials. Prog. Mater. Sci. 51 , 427–556 (2006).
33 S. Plimpton, Fast parallel algorithms for short-range molecular dynamics. J. Comput. Phys. 117 , 1–19 (1995).
34 G. Grochola, S. P. Russo, I. K. Snook, On fitting a gold embedded atom method potential using the force matching method. J. Chem. Phys. 123 , 7983 (2005).
35 Y. Mishin, D. Farkas, M. J. Mehl, D. A. Papaconstantopoulos, Interatomic potentials for monoatomic metals from experimental data and ab initio calculations. Phys. Rev. B 59 , 3393 (1999).
36 S. Alexander, Visualization and analysis of atomistic simulation data with OVITO–The open visualization tool. Modell. Simul. Mater. Sci. Eng. 18 , 015012 (2010).
37 J. J. Bean, K. P. Mckenna, Origin of differences in the excess volume of copper and nickel grain boundaries. Acta Mater. 110 , 246–257 (2016).
38 A. Zaïr , Effect of magnetism on the atomic structure and properties of Σ5 grain boundaries in fcc Fe and fcc Ni. Acta Mater. 226 , 117636 (2022).
39 D. Tsai, The virial theorem and stress calculation in molecular dynamics. J. Chem. Phys. 70 , 1375–1382 (1979).
40 Y. Cui, H. B. Chew, Machine-learning prediction of atomistic stress along grain boundaries. Acta Mater. 222 , 117387 (2022).
41 Q. Zhao , Imaging of atomic stress at grain boundaries based on machine learning. J. Mech. Phys. Solids 181 , 105455 (2023).
42 A. P. Bartók, R. Kondor, G. Csányi, On representing chemical environments. Phys. Rev. B 87 , 184115 (2013).
43 L. Himanen , DScribe: Library of descriptors for machine learning in materials science. Comput. Phys. Commun. 247 , 106949 (2020).
44 R. Lin, R. Zhang, C. Wang, X. Q. Yang, H. L. Xin, TEMImageNet training library and AtomSegNet deep-learning models for high-precision atom segmentation, localization, denoising, and deblurring of atomic-resolution images. Sci. Rep. 11 , 5386 (2021).33686158
