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J. Mater. Sci. Technol.  2020, Vol. 49 Issue (0): 236-250    DOI: 10.1016/j.jmst.2020.01.030
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Modes of grain growth and mechanism of dislocation reaction under applied biaxial strain: Atomistic and continuum modeling
Ying-Jun Gaoa,*(), Qian-Qian Denga, Zhe-yuan Liua, Zong-Ji Huanga, Yi-Xuan Lia, Zhi-Rong Luoa,b
a Guangxi Advanced Key Laboratory of Novel Energy Materials, Guangxi Key Laboratory for the Relativistic Astrophysics, School of Physics Science and Engineering, Guangxi University, Nanning, 530004, China
b Institute of Physics Science and Engineering Technology, Yulin Normal University, Yulin, 537000, China
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The phase field crystal method and Continuum Modeling are applied to study the cooperative dislocation motion of the grain boundary (GB) migration, the manner of the nucleation of the grain and of the grain growth in two dimensions (2D) under the deviatoric deformation at high temperature. Three types of the nucleation modes of new finding are observed by the phase field crystal simulation: The first mode of the nucleation is generated by the GB splitting into two sub-GBs; the second mode is of the reaction of the sub-GB dislocations, such as, the generation and annihilation of a pair of partial Frank sessile dislocation in 2D. The process can be considered as the nucleation of dynamic recrystallization; the third mode is caused by two oncoming rows of the dislocations of these sub-GBs, crossing and passing each other to form new gap which is the nucleation place of the new deformed grain. The research is shown that due to the nucleation of different modes the mechanism of the grain growth by means of the sub-GB migration is different, and therefore, the grain growth rates are also different. Under the deviatoric deformation of the applied biaxial strain, the grain growth is faster than that of the grain growth without external applied stress. It is observed that the cooperative dislocation motion of the GB migration under the deviatoric deformation accompanies with local plastic flow and the state of the stress of the system changes sharply. When the system is in the process of recrystallized grain growth, the system energy is in an unstable state due to the release of the strain energy to cause that the reverse movement of the plastic flow occurs. The area growth of the deformed grain is approximately proportional to the strain square and also to the time square. The rule of the time square of the deformed grain growth can also be deduced by establishing the continuum dynamic equation of the biaxial strain-driven migration of the GB. The copper metal is taken as an example of the calculation, and the obtained result is a good agreement with that of the experiment.

Key words:  Grain boundary splitting      Grain growth      Dislocation reaction      Atomistic simulation      Continuum modeling     
Received:  16 July 2019     
Corresponding Authors:  Ying-Jun Gao     E-mail:

Cite this article: 

Ying-Jun Gao, Qian-Qian Deng, Zhe-yuan Liu, Zong-Ji Huang, Yi-Xuan Li, Zhi-Rong Luo. Modes of grain growth and mechanism of dislocation reaction under applied biaxial strain: Atomistic and continuum modeling. J. Mater. Sci. Technol., 2020, 49(0): 236-250.

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Fig. 1.  Strain-driven small angle STGB migration and the GB splitting: (a) nanostructured bulk materials, (b) The magnified view of red box in Fig.(a), the red dashed line AB, CD, EF is the position of the initial GB. The slash line region indicates the region of the grain growth. (c) Simplified plot of rectangle ABFE regional enlargement of the GB [64] splitting and migration motion in the Fig.(b). It can be seen that the initial CD grain boundary dislocations are alternately arranged, in which there are two sets of dislocations, B2 and B3, or B5 and B6, and the GB splitting occurs under the biaxial strain. The “T” symbol in the figure indicates a dislocation Bi (where the description of Bi can be seen from the Appendix A), and the red dashed line indicates the initial position of the GB. The red dashed “T” indicates the dislocation arrangement of the initial GB. There is an angle between the Burgers vector of the dislocation and the GB direction. Yellow and green areas indicate the orientation of the grains. The direction of the exerted stress is shown in Fig. (c).
At low temperature (<0.6Tm) At high temperature (>0.6 Tm) Strain rate
Sample e ρ0 e ρ0 $\dot{ε}$
A [59] -0.30 -0.18 / / 6.0×10-
B / / -0.10 -0.195 7.2×10-6
Table 1  Parameters for sample preparation (Tm melting point).
Fig. 2.  The exerted biaxial strain on the sample: a tension is along x direction with εx, along y direction is a compression with εy, which results in a deviatoric deformation.
Fig. 3.  (a) The GB dislocation arrangement and the orientation of bicrystals. (b) The exploded view of the GB dislocation configuration : Each GB consists of three sets of dislocations (b2,b3,b4) or (b5,b6,b1), in which all dislocations are paired and arranged in a staggered arrangement; (c) The schematic of the arrangement and configuration of edge dislocation ({bi}) in the GB. Inset: the configuration of the GB dislocation pair Bi=bi+bi+1 is regarded as a combination of two Burgers vectors bi (partial dislocation). In the inset the dislocation b3 and b6 of the excess atomic plane along the grain boundary direction reflect the misorientation angle characteristic of the small angle STGB. (d) Free energy curve of the system vs time steps under the strain.
Fig. 4.  (a) Six possible atomic arrangement direction ${{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{i}}$ of a half of excess atomic plane of edge dislocation in 2D triangle lattice plane; (b) Six possible direction of Burgers vector ${{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{i}}$ of edge dislocation corresponding to the atomic arrangement direction${{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{i}}$. (c) Six possible directions of the composite vector Bi=bi+bi+1 and the directions of bi. The solid line arrow corresponds to Bi vectors, while the dotted short arrow corresponds to bi vectors. The direction of the Bi is rotated by 30° with respect to the direction of bi. The number in the Fig. 4(c) correspond to the order i in bi and Bi.
Fig. 5.  The evolution of the movement and interaction of the ATGB dislocations in the sample under biaxial strain. The direction of the yellow arrow in the figure indicates the direction of the slip of the dislocation pair; The white box area highlights the interaction of the edge dislocation pairs; The green band region “gap” indicates the deformed grain region; The yellow band region indicates the torsion region where the dislocations interact, i.e., the torsion zone. The strains for the figure are: (a) 0; (b) 0.0144; (c) 0.0316; (d) 0.0381; (e) 0.0432; (f) 0.0453; (g) 0.0497; (h) 0.0504; (i) 0.0533; (j) 0.0612; (k) 0.0633; (l) 0.0648; (m) 0.0684; (n) 0.0782; (o) 0.1008; (p) 0.1105; (q) 0.1152; (r) 0.1202; (s) 0.1235; (t) 0.1256; (u) 0.1307; (v) 0.1353.
Fig. 6.  The magnified image of the white box in Fig. 5(g) shows the reaction of the dislocation configuration. The dislocation pairs in the white box are approaching to annihilate. The red and yellow arrows indicate the direction of the dislocation movement. Annihilation of a yellow dislocation and a green dislocation occurs inside the white circle. The strain is: (a) 0.0504; (b) 0.0514; (c) 0.0540; (d) 0.0556; (e) 0.0633; (f) 0.0648.
Fig. 7.  The strain energy distribution of the sample during the process of the deformation is projected to x axis: OG: the region of the original grain; DG: the region of the deformed grain; RG: the region of recrystallization grain. The blue arrows in the figure indicate the direction of the strain energy gradients, and the green arrows indicate the direction of the extension and contraction of the strain energy platform. Strain: (a) 0; (b) 0.0144; (c) 0.0316; (d) 0.0381; (e) 0.0432; (f) 0.0453; (g) 0.0497; (h) 0.0504; (i) 0.0533; (j) 0.0612; (k) 0.0633; (l) 0.0648; (m) 0.0684; (n) 0.0782; (o) 0.1008; (p) 0.1105; (q) 0.1152; (r) 0.1202; (s) 0.1235; (t) 0.1256; (u) 0.1307.
Fig. 8.  (a) σx- εx curve, (b) σy-εy curve of the system under biaxial strain during the process of the deviatoric deformation, where the compression strain is in y direction and the tensile strain is in x direction. The stress σy and σy are just complementary.
Fig. 9.  (a) Schematic diagram of the initial GB dislocation structure of the grain system before the nucleation of the deformed grain. For the sake of the simplicity, the alternating arrangement of the composite dislocation vector Bi=bi+bi+1 here indicates the alternating arrangement of the GB dislocations, which structure of the GB is similar to that of Ref. [71]. (b) The alternating arrangement of dislocations Bi of the GBs splitting into two planar ATGBs shown in red box under the deviatoric deformation, and to generate new deformed grain 3 and 4 with orientation θ3 and θ4, respectively..
Fig. 10.  Curve of the growing area of the new grain of the PFC simulation changing with time driven by the biaxial strain. A is the growth area in grid numbers, and t is the time step(ts).
Stages: Grain: First stage of growth: Deformed grain with 0° orientation Second stage of growth: Recrystallization grain with 4° and -4° orientation Third stage of growth: Deformed grain with 0° orientation
The fitting formula: A-1.5=1.68×10-6×t1.89 A-1.5=7.6×10-6×t1.85 A-1.5=1.67×10-6×t1.91
fitting coefficient: α = 1.68 × 10-6, β = 1.89 α = 7.6 × 10-6, β = 1.85 α = 1.67 × 10-6, β = 1.91
Table 2  The fitting formula of grain growth of the PFC simulation in area.
Fig. A1.  (a) Schematic diagram shows the definition for the arrangement direction ${{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{i}}$ of a half of excess atomic plane of edge dislocation in 2D plane, and also for the direction of Burgers vector ${{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{i}}$ of the edge dislocation corresponding to the half of the excess atomic arrangement direction ${{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{i}}$: (I) for “up” direction of hi corresponding Burgers vector bi; (II) for “down” direction of hi corresponding Burgers vector bi; (III) The symbolic meaning of “⊥” of dislocation. (b) Schematic diagram of the six possible composite vector Bi of the dislocation pairs constituted by Bi=bi+bi+1, which corresponds to Fig. 4(c).
Fig. A2.  (a) Geometric representation of Burgers vector combination for Bi=bi+bi+1. (b) The calculation formula for the b1 and b2 vector combination:the parallelogram rule: (I) parallel situation, (II) vertical situation, (III) anti-parallel situation.
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