Topological Properties and Characteristics of Grain Boundary Structure in Metallic Materials

doi:10.11900/0412.1961.2025.00317

Acta Metall Sin

2026, Vol. 62

Issue (5): 705-720 DOI: 10.11900/0412.1961.2025.00317

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Topological Properties and Characteristics of Grain Boundary Structure in Metallic Materials

ZHU Siying(

), YI Min, GUO Wanlin(

)

State Key Lab of Mechanics and Control for Aerospace Structures, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Cite this article:

ZHU Siying, YI Min, GUO Wanlin. Topological Properties and Characteristics of Grain Boundary Structure in Metallic Materials. Acta Metall Sin, 2026, 62(5): 705-720.

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Abstract

The topological properties of metallic grain boundaries are crucial in determining their mechanical, electrical, and chemical behaviors, making them a major focus of grain boundary engineering. This study systematically reviews recent advancements in understanding the topological characteristics of metallic grain boundary structures at various scales, including atomic-scale topological configurations and mesoscale grain boundary network topology. It begins by summarizing current research on the topology of grain boundary atomic structures, including the coincidence site lattice model, displacement shift complete lattice theory, topological characterisation of grain boundary dislocation networks, and analysis of topological defects. It then introduces characterisation methods for mesoscale grain boundary networks, emphasising a research framework based on discrete cell complexes and systematically examining the topological properties of these networks. Finally, potential applications of grain boundary topology research in materials design are discussed.

Key words: metallic material grain boundary topology microstructure

Received: 10 October 2025

ZTFLH:

TG146

Fund: National Natural Science Foundation of China(T2293691);Fundamental Research Funds for the Central Universities(NJ2024001);Natural Science Foundation of Jiangsu Province(BK20243065);Natural Science Foundation of Jiangsu Province(BK20243044);Natural Science Foundation of Jiangsu Province(BK20251410);Research Fund of State Key Laboratory of Mechanics and Control for Aerospace Structures(MCAS-L-0525K01)

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	ZHU Siying
	YI Min
	GUO Wanlin

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https://www.ams.org.cn/EN/10.11900/0412.1961.2025.00317 OR https://www.ams.org.cn/EN/Y2026/V62/I5/705

Fig.1 Historical models for grain boundary structure^[3]
(a) amorphous cement model, uncrystallized atoms (black dots) fill the interstices between misoriented grains formed by rigid crystal units (rectangles)
(b) ordered boundary model, when rotated 36° around the [100] axis, one in every five atoms (black dots) at the interface is in a coincident position
(c) boundary transition region proposed, this model assumed that atoms (black circles) in the vicinity of the boundary would adopt relaxed positions

Fig.2 Grain boundary (GB) phase diagrams and their applications for materials design^[19] (T—temperature, T_M—melting temperature, X—solid solubility of Ga in Al, X_Max—maximal solid solubility of Ga in Al, MC—Monte Carlo method, MD—molecular dynamics. P and N—positive and negative transport channels, respectively)

Fig.3 Schematics (a, b) and high-resolution TEM (HRTEM) images (c-i) of GB plastic deformation initiated by excess volume through transient topological transition (TTT) (Scale bar: 2 nm)^[66]
(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 and blue circles denote atoms with excess volume above and below that of the perfect lattice (black circles), respectively. G1 and G2—grain 1 and grain 2)
(c-e) in situ HRTEM images showing TTT at general high-angle grain boundaries in Au with increasing compressive loading (Insets in Figs.3d and e are lattice strain images)
(c) atomic structure of a 24°

[1 1 ¯ 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 (yellow area) under compressive loading (σ) (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, g) atomic structures of the flat and curved segments of the initial 32°

[1 1 ¯ 0]

tilt GB
(h, i) TTT at the flat and curved GB segments under [112]_G1-compression (indicated by the black arrow)

Fig.4 Σ25 at 16° (a1), Σ13 at 22° (a2), Σ17 at 28° (a3), and Σ5 at 36° (a4) coincident site lattice configurations (reciprocal of coincidence site lattice (CSL) density Σ ≤ 25) obtained by rotations of the blue cubic lattice about a common [001] axis normal to the plane of the paper^[3] (Black lines indicate the CSL repeat units); schematics of Σ3 (b1) and Σ9 (b2) boundaries, strain-driven evolution of a Σ3 boundary (b3), and Σ3 ⁿ interaction (b4)^[104]

Fig.5 Schematics and high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) images of atomic configuration at tilt GBs in pure Ti^[2]
(a) three-dimensional (3D) atomic configuration and HAADF-STEM image of Fe “cages” at the Σ13 [0001] tilt GB (The GB plane is perpendicular to the figure, with basal planes of the adjoining crystals marked as A and B. Fe atoms (red spheres) periodically segregate between successive basal planes, forming icosahedral cage-like units along the [0001] tilt axis. These “cages” are surrounded by Ti (grey and blue spheres) pentagonal rings lying in the basal planes. The rotated Ti pentagons (blue and yellow dotted lines) reproduce the projected 10-atom ring along [0001])
(b1, b2) HAADF-STEM images showing atomic structure of symmetric Σ13 [0001] {

7 ¯

520} tilt GBs in pure Ti
(b1) the initial GB without detectable level of segregation (The periodic arrangement of structural units designated as A, B, and C)
(b2) periodic Fe segregation to a GB resulting in the formation of novel structures at the GB core resembling “cages” (The cage centre (red area) is rich in Fe, the shell of the cage (blue area) is rich in Ti, this result is also verified by atomic scale spectroscopy)

Fig.6 Topological grain boundary dislocation network transformations in {21 $3 ¯$ 0}[0001] orientation^[114]
(a) adding Ti atoms (light blue) to the left half of the ground-state structure and performing high-temperature molecular dynamics simulations trigger a dislocation-pairing transformation in a quasi-2D geometry. The top panels clearly show how the green dislocation lines pair up to form purple ones. The gray atoms are hcp coordinated, while red and dark blue atoms highlight different dislocation core structures
(b) analogously, adding Ti atoms (dark red) to the left half of the metastable structure triggers a dislocation-unpairing transition locally
(c) topological transition of the grain boundary dislocation network upon defect absorption. The view of the grain boundary plane shows a paired-dislocation grain boundary island (nucleus in purple) inside the parent ground state (green dislocation lines)

Fig.7 EBSD grain boundary maps (a-c), inverse pole figures overlapped with grain boundary maps (d-f), and corresponding misorientation angle distributions (g-i) of the hot-rolled titanium alloy sheet with different positions^[139] (RD—rolling direction, ND—normal direction)
(a, d, g) center (b, e, h) 2/5 of the 1/2 sheet thickness (c, f, i) 4/5 of the 1/2 sheet thickness

Fig.8 Mapping relationship between 3D grain boundary network and discrete cell complex structure (a)^[1], evolution ranges under different SPDs (SPD—severe plastic deformation, HAGBs—high angle grain boundaries, ARB—accumulative roll bonding, HPT—high-pressure torsion, ECAP—equal-channel angular pressing) (b)^[1], schematics of 3D grain boundary network evolution during CDRX (LAGB—low angle grain boundary) (c)^[153], and Betti number distributions through finite element method (FEM) sample (d)^[153]

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