Journal of Materials Science 【-逻*辑*与-】amp; Technology, 2020, 49(0): 236-250 doi: 10.1016/j.jmst.2020.01.030

Research Article

Modes of grain growth and mechanism of dislocation reaction under applied biaxial strain: Atomistic and continuum modeling

Ying-Jun Gao,a,*, 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

Corresponding authors: * E-mail address:gaoyj@gxu.edu.cn(Y.-J. Gao).

Received: 2019-07-16   Revised: 2019-08-30   Accepted: 2020-03-4   Online: 2020-07-15

Abstract

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.

Keywords: Grain boundary splitting ; Grain growth ; Dislocation reaction ; Atomistic simulation ; Continuum modeling

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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. Journal of Materials Science & Technology[J], 2020, 49(0): 236-250 doi:10.1016/j.jmst.2020.01.030

1. Introduction

Structure transformation of grain boundary dislocation (GBD) during plastic deformation process in nanometer- and submicro-sized polycrystalline materials are attracted great attention for decades [[1], [2], [3], [4]]. The change of the structure of the defect can greatly affect the mechanical properties of the materials. Many studies for sliding of the grain boundary (GB) have been done in polycrystalline materials. The research in recent years has been shown that the GB in nanostructure materials driven by the shear-coupled stresses moves in a way resembles to slip of the dislocation [[5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]]. This coupling can result in the strain-driven growth of grain in nanostructure materials and also affect the nucleation of the recrystallized grain. Two manners of nanograin growth [16,17] can be inspected. One is migration of the GB driven by the coupled shear, and the other is the rotation of the nanograin. The grain growth induced by the coupled stress in the existance of both solute segregation and configuration entropy [17] is also considered [18]. Now many researchers pay more attention not only to structural transitions [[19], [20], [21]] of the GBs, but also to grain growth under strain or stress at high temperature [[21], [22], [23], [24]]. For instances, the shear-coupled GB with the disconnection [18,20] and the transformation of the GBD from double dislocation pairs to disclination [24] under strain at high temperature. The GBD is movement in collective coordination under the strain, and the sub-GB which is split from the GB with opposite sign of the Burgers vector of the dislocation meets each other to annihilate [25,23], and even the Burgers vector of the dislocation occurs to rotation [26] and reaction of the dislocation in the GB [27]. The effect of the stress coupling the GBD causes the GBD movement in two ways [[28], [29], [30], [31], [32], [33], [34]]. One is the dislocation along the direction of the GB sliding, and the other is the dislocation leaving from the original GB position and sliding along the vertical direction of the GB, in which the GBD is in collective migration. Whether it is the first way of the collective sliding or the second way of the collective migration, it depends on the structure and configuration of the GBD [29], one of which is the discrete dislocation model with a sessile extrinsic grain boundary dislocation (EGBD) array [30], and the other of which is the model with a glissile EGBD array [30]. Usually the GBD with the sessile EGBD array coupling to the applied shear stress is moved in gliding along the GB, while the GBD with the glissile EGBD array coupling to applied shear stress is moved in migration leaving from the original GB position [29,35]. The stress-driven migration of the GBs contributes to both plastic flow and grain growth in materials [28]. Although, very recently, the grain growth through splitting GB to migrate under a shearing stress in nanomaterials is reported [30,31], so far, the biaxial strain-driven GB splitting [25,36] or the dislocation configuration exchange of the encountering SGBs in opposites [37] at high temperature is still unclear.

Even though molecular-dynamic (MD) [[37], [38], [39], [40], [41]] has been successful in studying the movement of the GBD and the migration of the grain boundary under the stress, an obvious limitation of the MD method is that its time scale is generally limited to nanoseconds (10-9s) [42]. As a consequence, only high stress (GPa) and stress rates (107s-1) can be probed in dynamical deformation processes and is of diverge several orders from the actual experiment owing to the limited time scale accessible to MD simulations. Traditional phase field (TPF) can be also applied to simulate grain growth. However, it is difficult for the TPF to show the dislocation of the GB [[43], [44], [45]]. Based on density functional theory, new atomistic method named as phase field crystal (PFC) [46,48] is proposed in 2002. Not only the PFC simulation overcomes the limit of the time scale of the MD, but also the applied strain rate in the PFC simulation is consistent with the experimental results. Therefore, the PFC model has been successfully brought into play to many fields [[48], [49], [50], [51], [52], [53], [54], [55]] of materials since it proposed. The effects of previous research [[9], [10], [11], [12], [13]] on the discussed area were focused mostly on coordinative dislocation movement (CDM) of the GB and on continuously GB evolution (that does not affect GB misorientation) of nanomaterials under the stress.

Recent researches [25,26,30,32,37] indicates that the dislocation motion at the nanograin boundary under loading at high temperature is very different from that at normal temperature, and many new phenomena have emerged. For example, the pairing and splitting of dislocations, proliferation and decomposition of the dislocation pair, separation and rotation of the pair, synergistic motion of the GB dislocations and the GB splitting, and so on. The evolution of these dislocation configurations strongly depends on the mode and magnitude of the applied strain at high temperature. At present, the research in these area is still not deep enough, and only very few researches [25,[56], [57], [58], [59]] concentrated on the DCM of the GBD and on the biaxial strain-driven GB splitting into two new GBs [37] or on the dislocation configuration permutation [14] of the encountering SGB. Based on the PFC model [26,60] for the GBD, the main aim of this paper is that we study the CDM of the GB migration and the modes of the nucleation and growth of grains under deformation at high temperature, and also study the localized strain energy under the biaxial strain by the PFC model. Furthermore, we reveal the mechanism of the GB splitting into two groups of SGBs and also the permutation of the dislocation configuration of the SGBs on nanoscale, and deeply understand the growth law for the deformed grain under the strain by the continuum model of the GB migration.

2. Model and method

2.1. PFC model

The free energy functional $\overset{}{\mathop{F}}\,$ of the system in PFC model given in Refs. [46,47] is as

$\overset{}{\mathop{F}}\,=\int f(\rho (x(1+\varepsilon ),y(1-\varepsilon )))dV=\int [f(\rho (x,y))+{{E}_{ext}}(\varepsilon ,x,y)]dV$

where the local free energy f(ρ) is as

$f(\rho (x,y))=\left[ \frac{1}{2}e{{\rho }^{2}}+\frac{1}{4}{{\rho }^{4}}+\frac{1}{2}\rho {{(1+{{\nabla }^{2}})}^{2}}\rho \right]$

where ρ is an order parameter corresponding to atomic density and e is a temperature parameter, and ∇2 the Laplace operator. Eext is the changed energy by external force, and is written as

${{E}_{ext}}(\varepsilon ,x,y)={{V}_{ext}}\cdot \rho$

where Vext is external force, and the detailed expression can be seen in Ref. [26]. The atomic density ρ is obtained in the one-mode approximation [46,60] for the free energy, which can be expressed for the triangular structure two-dimensions as

$\rho (x,y)={{A}_{T}}[\cos (qx)\cos (qy/\sqrt{3})-\cos (2qy/\sqrt{3})/2]+{{\rho }_{0}}$

Where $q=\sqrt{3}/2$ the wave vector and ρ0 is the average atomic density.

The atomic density evolution of the system is governed by the Cahn-Hilliard equation [46,60]

$\frac{\partial \rho }{\partial t}={{\nabla }^{2}}\frac{\delta \overset{}{\mathop{F}}\,}{\delta \rho }={{\nabla }^{2}}\left[ e\rho +{{\rho }^{3}}+{{(1+{{\nabla }^{2}})}^{2}}\rho +{{V}_{ext}} \right]$

where the detail of Vext is given in Ref. [26,51,59]. The evolution of the atomic density in 2D is governed by Eq. (5). The atomic density can be gotten by making use of the semi-implicit Fourier spectral method to solve the Eq. (5) [61,62].

2.2. Sample for simulation

The GBs in nanocrystalline metals play an important role of sources and sinks of dislocation, and the migration and slip of the GB extremely effect the rotation and growth of the nanocrystals. In order to fully understand the role of the strain-driven GB migration in both plastic deformation and local deformed grain extension in nanocrystals, it is very important to reveal the mechanism of structure transformations of the GB. Fig. 1 shows the schematic of the strain-driven GB splitting and migrating in nanocrystallione materials and also the transformation of the GB misorientation [[30], [31], [32], [33]]. Thus, the strain-driven GB migration not only is localized within one grain, but also generates new migration case [33] and even makes neighboring grains grow [30,32].

Fig. 1.

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).


Many examples [[24], [25], [26], [27], [28], [29], [30]] of the structural transformation and the migration of the GBs in bicrystal structure systems are studied under the applied strain. In this work we focus on the GB splitting and the synergistic migrating of the GB dislocations, and the dislocation crossing between these GBs. For simplicity, here we ignore the generation of the disclination [31] caused by the coordinated migration of the GB and also the influence of the triple junction of the GBs on migration of the GB during the deformation process under the strain. In this case, we here only consider the bicrystal system with small angular symmetric tilt grain boundaries (STGBs) under the action of applied biaxial strain, in order that the GB splits into two straight SGBs, and these SGBs meet each other to result in the dislocation reactions and its configuration change [37], all of which have a significant effect on new grain nucleation and grain growth. Therefore, the migration and splitting of the GB is regarded as pure strain-driven GB. The bicrystal system here can be easily used to highlight the dislocation cross-slipping that reflects the migration and the encounter and the splitting of the GB, and also to show the details of the interaction between the dislocations. For example, the proliferation, annihilation, climb, slip of the dislocation, and the orientation change of the grain during the migration of the GB. For the PFC simulations of the GB migration and of the cross-slipping of the dislocation encountering with the oncoming GB dislocations under the strain in Fig. 1(c), the periodic boundary conditions and equal-area deformation assumptions [57,62] are used.

Based on three reasons presented in Ref. [37], the migration and splitting of the GB in a simplified two-dimensional system is studied in the present paper, which is also suitable for the study of grain boundary migration and grain growth in multilayer film grain system [63]. The two-dimensional atomic lattice structure of the sample of the numerical simulation experiment [26,27] designed in this paper is similar to the lattice structure of the (111) plane of the fcc crystals. Here we choose the sample with a square area ${{L}_{x}}\times {{L}_{y}}=1024\Delta x\times 1024\Delta y$, which scale is about 60*60 (nm)2, for simulating. The simulation system (hereinafter referred to as the sample) is set in bicrystal structure [27,48,63]. The parameters given in the PFC phase diagram [47] for the sample preparation are shown in Table 1.

Table 1   Parameters for sample preparation (Tm melting point).

At low temperature (<0.6Tm)At high temperature (>0.6 Tm)Strain rate
Sampleeρ0eρ0$\dot{ε}$
A [59]-0.30 -0.18/ / 6.0×10-
B/ /-0.10 -0.195 7.2×10-6

New window| CSV


In order to obtain the sample with the small-angle STGB, following the procedure in Ref. [26], we can design it by using the formula (4) of the atomic density distribution [25,26]. Here the misorientation of the STGB is θ = 8° (it can be set one half of the domain is -4° and the other half 4°). Here the real temperature [26] T corresponding to e = 0.10 is about T > 0.6 Tm for Cu fcc metal, which indicates that the sample is in high temperature.

2.3. Applied strain and related formulas

For simplicity, the applying strain method in the PFC model can be easily and directly used under constant volume condition [63] to avoid designing a complex compensation function for boundary condition [42]. Since a constant strain rate is applied to all atoms, the deformation state becomes the affine deformation state, and periodic boundary conditions can be easily used as boundary conditions [63].

Following Ref. [58] for the tensile deformation, we carry out the deformation simulation for sample B by using a deviatoric deformation condition [58] under the biaxial strains. The biaxial strain rate is set to be a constant $\dot{\varepsilon }=7.2\times {{10}^{-6}}/\Delta t$. The strain has the form of $\varepsilon ={{\varepsilon }_{x}}={{\varepsilon }_{y}}=\dot{\varepsilon }n\Delta t$, where Δt is time step (ts) and n is the number of the time step. Under the strain, we have

$\begin{array}{*{35}{l}} \Delta x\cdot \Delta y=\Delta {{x}_{0}}(1+{{\varepsilon }_{x}})\cdot \Delta {{y}_{0}}(1+{{\varepsilon }_{y}}) \\ =\Delta {{x}_{0}}\cdot \Delta {{y}_{0}}(1+{{\varepsilon }_{x}}+{{\varepsilon }_{y}}+{{\varepsilon }_{x}}{{\varepsilon }_{y}})=\Delta {{x}_{0}}\cdot \Delta {{y}_{0}} \\ \end{array}$

Where Δx0 and Δy0 are the initial grid sizes, Δx and Δy are the deformed grid sizes at the nth time steps of the PFC simulation. We use the constant area condition [58], i.e., $S=\Delta {{x}_{0}}\Delta {{y}_{0}}$=$\Delta x\Delta y$during the deviatoric deformation, more details of which can be seen in Ref. [57,62], and we can get

$(1+{{\varepsilon }_{x}}+{{\varepsilon }_{y}}+{{\varepsilon }_{x}}\cdot {{\varepsilon }_{y}})=1$

where εx and εy is a small amount of the strain, i.e, εx and εy <<1, then εx×εy is the second order small amount, and can be ignored. Then we have εx + εy = 0, i.e, εx = -εy. Under this condition of the deformation, the deviatoric deformation exerted on the sample with a constant area occurs in two dimensions. For a plane strain, we here have the relationship [65,66] between the strain and stress

${{\varepsilon }_{x}}=\frac{1-{{\nu }^{2}}}{E}({{\sigma }_{x}}-\frac{\nu }{1-\nu }{{\sigma }_{y}})$
${{\varepsilon }_{y}}=\frac{1-{{\nu }^{2}}}{E}({{\sigma }_{y}}-\frac{\nu }{1-\nu }{{\sigma }_{x}})$

where ν is the Poisson’s ratio, and E is the Young’s modulus. When satisfy εx =ε =-εy, we can have σx = -σy from these formula above, then we get ${{\varepsilon }_{x}}=\frac{1+\nu }{E}{{\sigma }_{x}}$ and ${{\varepsilon }_{y}}=-\frac{1+\nu }{E}{{\sigma }_{x}}$, or ${{\sigma }_{x}}=\frac{E}{1+\nu }{{\varepsilon }_{x}}$ and ${{\sigma }_{y}}=-{{\sigma }_{x}}=-\frac{E}{1+\nu }{{\varepsilon }_{x}}$.

As the bicrystalline sample is exerted under the deviatoric deformation, i.e., the sample is pulled along x direction and is extruded along y direction as shown in Fig. 2. To determine the mechanical properties during the deviatoric deformation process, we calculate the strain energy on the sample as: $F=F(\varepsilon )-F(\varepsilon =0)$, where $\varepsilon ={{\varepsilon }_{y}}=({{L}_{y}}-L_{y}^{0})/L_{y}^{0}$ is the engineering strain applied in the y direction, The engineering stress [48,49] can be written as:

${{\sigma }_{y}}=\frac{1}{{{A}_{0}}}\frac{\partial F}{\partial {{\varepsilon }_{y}}}=\frac{1}{{{{\dot{\varepsilon }}}_{y}}}\frac{\partial (F/{{A}_{0}})}{\partial t}$

where ${{A}_{0}}=L_{x}^{0}\times L_{y}^{0}$ is the initial size, and $F/{{A}_{0}}$ is the free energy including the strain energy. Similarly, the engineering strain in the x direction is obtained as

${{\sigma }_{x}}=\frac{dF}{d{{\varepsilon }_{x}}}\text{and}{{\sigma }_{y}}=\frac{dF}{d{{\varepsilon }_{y}}}=-\frac{dF}{d{{\varepsilon }_{x}}}=-{{\sigma }_{x}}$

where σx and σy is the average normal stress.

Fig. 2.

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.


The elastic properties of the triangular lattice in 2D can be obtained by considering the energy costs for deviating equilibrium state. The free-energy density associated with deviatoric deformation can be calculated by considering modified forms of Eq. (4), i.e., $\rho (x(1+\varepsilon ),y(1-\varepsilon ))$. In such calculation the strain εrepresents the dimensionless deformation. The results of the calculations to determine the elastic constants and strain energy for the 2D system given in Ref. [48] is as below

${{F}_{dev}}={{F}^{\min }}+{{F}_{el}}={{F}^{\min }}+[{{C}_{11}}-{{C}_{12}}]\cdot {{\varepsilon }^{2}}$

The elastic constants are then

${{C}_{11}}/3={{C}_{12}}={{C}_{44}}=\alpha /4$

These results are consistent with the symmetries of the 2D triangular system, i.e., ${{C}_{11}}={{C}_{12}}+2{{C}_{44}}$· For the bicrystal with two flat STGBs without curvature, when the deviatoric deformation is applied to the simulation cell, it results in a deviatoric strain to drive the GB migration. The driving force is equal to ${{f}_{d}}=\Delta {{F}_{dev}}={{E}_{str}}=m({{C}_{11}}-{{C}_{12}})\cdot {{\varepsilon }^{2}}$ [68,67], where m is a constant.

3. Result and discussion

3.1. Structure of GB dislocation in simple B

It has been reported in Ref. [37] for sample A that the GB dislocations have undergone a simple migration and annihilation at room temperature without exchanging configuration. Here we concentrate on sample B at high temperature. The prepared sample B with the small-angle STGB is of a bicrystal structure with orientation angle 4°and -4°as shown in Fig. 3(a), in which the array direction of the GB dislocation is along the [0,1] direction of y axis shown in Fig. 4. In order to more clearly display the dislocation structure of the GB in Fig. 3(a)-(c) show the sketch map of the lattice dislocation arrangement of the two GBs. Fig. 3(b) shows that the schematic diagram of the dissociated arrangement of the dislocations of one GB, in which the GB consists of three sets of dislocations (b2,b3,b4) or (b5,b6,b1) owing to the STGB. The vertical arrangement of the dislocations b6 in Fig. 3(b) constitutes a small-angle symmetric tilting GB, and the tilting angle θ is θ=b/D1 = 8°, where b is the amount of Burgers vector and D1 is the spacing of the arrangement of the dislocation b6. The spacing D2 of the tilting arrangement of the dislocations (b5 or b1) is D2≈2D1, and each of the arrangement forms a small angle asymmetric tilt grain boundaries (ATGB) [64] and the tilting angle is ${{\theta }_{2}}=\pm b/{{D}_{2}}\pm b/(2{{D}_{1}})=\pm \frac{\theta }{2}=4\circ $ as shown in Fig. 3(b) and (c). In some interface structures, a secondary dislocation structure also occurs [65].

Fig. 3.

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.

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.


The schematic map of the arrangement and configuration of the dislocation ({bi}, {Bi}) in the GB is shown in Fig. 3(c). All dislocations are paired and arranged in a staggered arrangement. The structure of the GB dislocation pair has been observed in Fig. 5 of Ref. [68]. It can be seen that the GB is composed of eight lattice dislocation pairs, which is represented by the symbol Bi= bi+bi+1 in Appendix A, where a single dislocation bi cannot independently exist and can be regarded as a partial dislocation [69]. The more details of Bi and bi are explained in the Appendix A. It can be seen in the same STGB of Fig. 3(c), that the direction of the Burgers vector of the closest lattice dislocation pair [69] is different, and the angle difference of their Burgers vectors is about 60°, while for the interval lattice dislocation pair [37,69], the Burgers vector directions are in the same. It follows that each GB is constituted of two types of the dislocation pair marked as I and II (B5 and B6) or the III and IV (B2 and B3) types, while the Burgers vectors of the GB dislocations in different column are in opposite. Therefore, there are four kinds of the lattice dislocation pairs Bi in the two GBs as shown in Fig. 3(c).

Fig. 5.

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.


As the Ref. [61] pointing out, for two dimension triangular lattice, the lattice edge dislocation vector is only of six possible orientation direction, in which there are only two basic Burgers vectors, and all the six direction vectors can be expressed by the two basic Burgers vectors. Fig. 4(b) shows the six possible directions of the Burgers vector of the edge dislocation and Fig. 4(a) shows them corresponding to the six directions hi of the half extra atom array of the dislocation in 2D lattice. Fig. 4(c) shows the six possible directions of the composite vector Bi=bi+bi+1 and the directions of vectors bi. For example, the six Burgers vectors bi in Fig. 4(b) are expressed in 2D direction index

${{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{1}}=[-\frac{1}{2},\frac{\sqrt{3}}{2}]{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{2}}=[\frac{1}{2},\frac{\sqrt{3}}{2}]{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{3}}=[1,0]{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{4}}=[\frac{1}{2},-\frac{\sqrt{3}}{2}]{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{5}}=[-\frac{1}{2},-\frac{\sqrt{3}}{2}]{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{6}}=[-1,0]$

where the lattice spacing a is set as 1. The detailed description of the relationship among hi, bi and Bi can be seen in the Appendix A.

3.2. Mechanism of dislocation reaction of sub-GB exchange and intersection under biaxial strain

Fig. 5 shows the movement of the GB dislocations in the sample under deviatoric deformation. The whole process of the migration of the sub-GB, from the system energy point of view, can be divided into six major stages, as shown in Fig. 3(d). Under the biaxial strain, the STGB splits into two sub-GBs, which can be regards as splits into two asymmetric tilt grain boundaries (ATGB), for example, the ATGB I and II, or III and IV as shown in Fig. 5(a)-(d). The separation of the GB dislocation creates gaps [34] between the two split ATGBs. The gap can be considered as a new generated grain with 0°orientation as shown in Fig. 5(d) and (e). Under the strain, these ATGB dislocations slide in cooperative movement in a straight row, and leaves away from the two sites of the original GB. It is worth noting that the ATGB I and IV, or III and II approach and meet each other face to face, and form the dipole of the edge dislocation pair in a temporary stability, as shown in Fig. 5(g). Then, the configuration exchange of two columns of the dislocations of the ATGB occurs. For example, the ATGB I and IV exchanges into the ATGB II and III, and the ATGB II and III exchanges into the ATGB I and IV, as shown in Fig. 5(f)-(m). In order to observe more carefully the variation characteristics of the configuration of the dipole of the edge dislocation and of the dislocation reaction in the process under the biaxial strain, here we choose the dipole of the dislocation pair to analyze in the white rectangle box in Fig. 5(g), where two dislocation pairs are moving toward each other. The detail interaction process of the dipole of the dislocation pair inside the white rectangle box with the increase of the strain is shown in Fig. 6(a)-(f). It can be seen in Fig. 6(a) that these two dislocation pairs attract and approach each other to form a dipole of the dislocation pair and also to form a distortion zone around the dislocation, which approaches each other and gradually connects, as shown in Fig. 6(b). By observing carefully, we discover that one yellow and one green dislocation seem to form a pair of partial Frank sessile dislocations in 2D, which is similar to a loop of Frank sessile dislocation [67,69,64] in three dimensions. There is a red color vacancy layer flatted between the yellow and green dislocation pairs shown in Fig. 6(b), i.e, the pair of Frank sessile dislocation with a collapsed vacancy disc. Under the effect of the compressive component of the strain which is resolved along the axis direction of the pair of the Frank sessile dislocation, the pair of Frank sessile dislocations have to climb and approach to each other [64,70,71], and to make the collapsed vacancy disc shrinking as shown in Fig. 6(a)-(c). Then, one yellow and one green Frank sessile dislocations approach and meet each other to annihilate inside the white circle shown in Fig. 6(c). At this time another yellow and green dislocations only remain in the region. The two original local distortion zones merge and change into an elliptical region shown in Fig. 6(d). As the amount of the strain continues increasing, a pair of new purple dislocation initiates in the place where there are only one yellow and green dislocations remained, as shown in Fig. 6(d). The new purple dislocation initiation makes the free energy of the system increase and reach maximum at the point k, as shown in Fig. 3(d). In this case, it can be seen that its Burgers vector direction has rotated about clockwise 60° in Fig. 6(b) and (e). Then, the new generated purple dislocation glides along the direction of the shearing component of the strain, shown in Fig. 6(e). Finally, the dipole of the dislocation pair decomposes and separates, and glides away in the opposite direction, as shown in Fig. 6(f). According to the Fig. 4(b), the Burgers vectors of yellow, purple, green dislocation pairs shown in Fig. 6 can be expressed as below:

Fig. 6.

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.


Yellow dislocation ${{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{3}}=[1,0]$, ${{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{5}}=[-\frac{1}{2},-\frac{\sqrt{3}}{2}]$;Green dislocation ${{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{2}}=[\frac{1}{2},\frac{\sqrt{3}}{2}]$, ${{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{6}}=[-1,0]$; Purple dislocation ${{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{1}}=[-\frac{1}{2},\frac{\sqrt{3}}{2}]$ ${{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{4}}=[\frac{1}{2},-\frac{\sqrt{3}}{2}]$;

The reaction processes of the dipoles of the dislocation pairs in two stages for nucleation of new grain during the ATGBs exchanging are shown as below:

a)${{B}_{2}}+{{B}_{5}}\to \left( {{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{3}}+{{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{2}} \right)+\left( {{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{6}}+{{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{5}} \right)\to {{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{3}}+\left( {{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{5}}+{{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{2}} \right)+{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{6}}\to \left( {{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{3}}+\underset{\scriptscriptstyle-}{0}+{{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{6}} \right)$, where $\left( {{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{5}}+{{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{2}} \right)$ is the formation of a pair of Frank sessile dislocation, and ${{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{5}}+{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{2}}=\underset{\scriptscriptstyle-}{0}$ is the annihilation of the pair of one yellow and green Frank sessile dislocation by climbing with the vacancy layer flatted.

b)$\left( {{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{3}}+\underset{\scriptscriptstyle-}{0}+{{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{6}} \right)\to {{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{3}}+\left( {{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{1}}+{{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{4}} \right)+{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{6}}=\left( {{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{1}}+{{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{6}} \right)+\left( {{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{4}}+{{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{3}} \right)={{B}_{3}}+{{B}_{6}}$), where 0 ⇒$\left( {{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{1}}+{{{\overset{\scriptscriptstyle\rightharpoonup}{b}}}_{4}} \right)$, is the generation of the two purple dislocations by the action of the shearing component of the strain. Therefore, the final result of the dislocation reaction is B2+B5B3+B6, in which the configuration and direction of the dislocation B2 and B5 have changed into B3 and B6.

After the transformation process of the configuration of the dislocation of the ATGBs finishes, the ATGBs return and go back in the initial path in Fig. 5(l)-(n). In the return process, when the dislocations of the ATGBs meet again in the manner of the intersection of the slipping line, they do not form the dipole of the dislocation pair or the locked structure [69,65,[70], [71], [72]], but take an avoidance mechanism [34] of the dislocation to find a way to glide and pass each other as shown in Fig. 5(m)-(n), and the dislocation arrangement of the ATGB is restored to the arrangement structure of the original GB shown in Fig. 5(n). At this time, the energy of the system reaches the lowest point n in Fig. 3(d), which means that these two ATGBs merge and restore into the initial GB state. After that, the pair of the dislocations with opposite Burgers vectors approaches and meets each other in the second time in Fig. 5(q)-(t). They do not form the relative stable structure of the dipole of the dislocation pair, but annihilate directly and vanish as shown in Fig. 5(t)-(v).

3.3. Profile and propagation of strain energy

The evolution of the strain energy profile of the sample by projecting the strain energy to x axis are shown in Fig. 7(a)-(u), which is a sum of the strain energy density values over y-axis plotted as a function of x. The strain energy profile without the applied strain is given in Fig. 7(a). It can be seen that the energy at the GB position is about 0.0185 higher than that about 0.0170 inside the grain. With the applied strain increasing, the peak of the energy profile at the GB rises up and becomes widening, as shown in Fig. 7(b)-(e). This indicates that the strain energy is firstly concentrated on the GB. The widened peak of the energy profile indicates that the GB dislocation is divided into two columns to glide away from the two side of the original GB. This collective cooperative sliding motion of the dislocations causes local plastic flow.

Fig. 7.

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.


As the amount of the strain continues increasing and reaches 0.0432, the strain energy stored in the GB reaches the maximum about 0.028 as shown in Fig. 7(e). From that time on, with the strain increasing, the profile shape of the strain energy in the GB changes from the narrow peak to the widened platform DG as shown in Fig. 7(e), where the platform DG corresponds to the “gap” area with green slash area in Fig. 5(d) and (e). However, the height of the peak remains a constant 0.028 without rising, as shown in Fig. 7(f). This indicates that these dislocations collectively and cooperatively glide away from the GB to form the local plastic flow [[73], [74], [75]]. The strain energy will increase in the place swept by the plastic flow, which area is the new deformed grain here referred to as the deformed grain (DG) with higher strain energy shown in Fig. 7(f)-(h). The width DG of the platform of the energy profile corresponds to the width of the DG. In this process of the platform extension, we consider that it occurs that the higher energy region extends into the lower energy region [76] of the original grain (OG). When the two columns of the local plastic flow propagates toward and meets each other, as shown in Fig. 7(e)-(h), the interaction between these dislocations occurs in Fig. 5(h)-(k). It can be seen from Fig. 7(f)-(g), that two DG platforms meet to form groove structure, in which the height of the energy cannot reach the height of the energy platform. This indicates that the two columns of the dislocations interact with each other to annihilate by one yellow and green Frank sessile dislocations in Fig. 6(c) to release a part of the strain energy. Therefore, the energy density of the region cannot reach the maximum height 0.028 of the platform. After that, the platform width DG of the energy profile starts to shrink as shown in Fig. 7(g)-(j), which indicates that the two columns of the dislocations start to go back in the opposite direction. This corresponds to the growth of the recrystallization grain (RG) in Fig. 5(k)-(m). During this process, the dislocation glides and releases the strain energy accumulated in the deformed grain, which can be considered as the new lower energy region extending into the higher energy region [76], i.e., the width RG of the recrystallization grain extends into the DG region of the deformed grain. Finally, the distribution of the strain energy restores to the initial state in Fig. 7(n), similar to Fig. 7(a). It can be considered that the elastic deformation dominates during the deformation process, because the deformed state of the system can be fully restored to the initial state [76], and also the strain energy stored in the DG can be completely released. With continuing to apply the strain, the accumulation of the strain energy of the GB dislocation can occur again, as shown in Fig. 7(n)-(o). When the stored strain energy at the GB reaches a maximum again, the peak of the energy curve keeps the constant shown in Fig. 7(o)-(p), and also the width of the peak only extends to both sides to form the platform DG again, as shown in Fig. 7(p)-(r). The two DG platforms meeting each other mean that the DG with high strain energy merge and the RG region shrinks and disappears. At the same time, the two column dislocations meet each other to annihilate, rather than that the movement of the dislocation turns it direction and goes back again. During the process of the dislocation annihilation, the strain energy of the system does not release completely in Fig. 7(p)-(t). This stage is similar to the first stage in which the higher energy region extends into the lower energy region. It also can be seen in Fig. 7(t)-(u) that the distribution of the strain energy in the system reaches the uniform distribution and is higher than that of the initial distribution.

It seems intuitive [76] that the strain energy would be reduced by the motion of the GBs towards the grain with higher energy, however, the simulation results of this works for the bicrystal under the biaxial strain shows that the ATGB moves towards the OG with lower strain energy. The similar phenomenon is also presented by Tonks et al [75]. In addition, it is found that the deformed grain (DG) is softened by the deformation [33,[76], [77], [78]]. The softened DG, due to the plastic flow localization (i.e, the collective cooperation movement of the dislocations driven by the strain), can store more strain energy because it accommodates more of the applied strain than the OG [33,75]. The plastic flow localization usually occurs in the process of the deformation of nanocrystalline (NC) metal [73] and metallic glass [[76], [77], [78], [79]], in which involves local nonlinear physical process [[79], [80], [81]].

3.4. Stress state and plastic flow

Materials yield and deform plastically under external applied strain. On atomic scales, the external strain is carried by localized crystal defects, for example, dislocations and disclinations [82]. They are created under the strain and interact with each other. A successful theory of plasticity is often linking the microscopic behavior of individual discrete dislocations to the macroscopic plastic behavior of the system [84]. The plastic flow in periodic systems is typically mediated by the motion of the dislocations. Although the properties of individual defects and their interactions are well known, their collective behavior under external stress is complicated, and it gives rise to scaling-invariant phenomena [84]. These phenomena include dislocation slip avalanches [85]. Because the PFC model captures the nonlinear elastic behavior of a crystal, the interaction between dislocations is naturally captured. In addition, because the PFC model simulates the atoms in the lattice but not the dislocations themselves, creation and annihilation of them are also naturally captured as collective excitation of the lattice [85]. No ad hoc regulations or assumptions have to be forced [85]. Because the manner of the biaxial strain application in this paper is different from the usual manner of the applied tensile stress, the stress-strain (S-S) curve linking to the dislocation motion in this paper is more complex.

Here Fig. 8(a) and (b) show the S—S curve of the whole process of the deviatoric deformation in the x and y directions under the applied biaxial strain, respectively, i.e, σx-ε and σy-ε curves. The two curves are just complementary under the constant area condition. In a-b section of the σx-ε curve in the range of the strain 0~1.44% in Fig. 8(a), the deformation is the elastic deformation owing to the fact that the stress increases linearly with the strain. In this process, the elastic strain energy of the system monotonically increases with the strain. This is owing to the dislocations of the GB accumulating strain energy on it. The yield point εp for the stress σx is at b. In b-c section of the strain 1.44~3.16% in the σx-ε curve, the dislocations begin to slip collectively and cooperatively to form plastic flow, and the plastic deformation occurs. During the process, the increase of the stress becomes slowly and reaches a positive maximum point c as shown in Fig. 8(a). In c-d-e section of the σx-ε curve in the strain 3.16~4.32%, although the energy of system still increases and reaches a maximum at point d in Fig. 3(d), the stress σx begins decreasing until to reach point e in Fig. 8(a), which case corresponds to the strain softening characteristic [76]. According to Ref. [[76], [77], [78], [79], [80], [81]], the formation of the localization deformed band DG shown in Fig. 7(d) can be considered as the reason for the strain softening.

Fig. 8.

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.


In e-f section of the curve in the strain 4.32~4.53%, the energy of the system still decrease in Fig. 3(d), while the stress σx begins to increase from the minimum point e to reaches the maximum at point f with the strain 4.53% shown in Fig. 8(a). In f-g-h-i stage of the curve in the strain 4.53~5.33%, the stress σx of the system deceases from the maximum f down to the minimum h, which corresponds to that the Frank sessile dislocation pairs attract and annihilate to reduce dislocation numbers as shown in Fig. 5(f)-(i). During this process, despite the strain exerts on the sample, there is still local stress relaxation inside the system which is owing to the fact that there are intense local synergistic movement and attraction between these diploe dislocations inside the system [[83], [84], [85]]. In the whole d- e-f -g-h-i stage, the stress σx state of the system changes dramatically.

In the i-j-k process in the strain 5.33~6.33% of the curve, the curve of the stress σx is in a concave shape, i.e., the stress firstly increases to position maximum point j and then decreases to point k. At this time, the purple dislocation pairs generate in Fig. 6(d), and the dipole dislocation pair approaches and squeezes each other to make the local dislocation number increase and also makes the energy of the system increase (seeing Fig. 3(d)). The added energy of the system in this stage is the energy for nucleation of recrystallization grain. While in the k-l-m-n stage of the strain 6.33~7.82% in the σx-ε curve, the stress is also in lower stress state in Fig. 8(a) under the biaxial strain, this indicates that the energy of system is in a instable state and in relaxation from maximum point k quickly to minimum point m in Fig. 3(d) which stage of the released energy corresponds to the grain growth of dynamic recrystallization. Due to the collective and cooperative sliding motion of the dislocations in the recrystallization grain growth, the strain energy stored inside the dislocations is released simultaneously. This means that the plastic flow propagates reversely along the direction of the gradient of the strain energy in the GB as shown in Fig. 7(l). During this stage, the strain energy spreads from the low-energy region into the high-energy region to reduce the system energy. The literature [31,86] also reports that there will be a reversed plastic flow phenomenon occurs in the state of negative stress. This may be owing to the instability of the localized plastic deformation which makes the DG transform into OG to restore its orientation [[79], [80], [81], [82],87].

In the N-1-o-p stage of the strain 7.82~11.05% in the σx-ε curve, the stress is in high positive state. During this stage the stress σx firstly increases to the maximum at point o and then decreases to point p, which corresponds to that the dislocations of two grain boundaries intersect by cross-slipping of the ATGB dislocation and also the deformed grain grows in second time. While in p-q-r stage of the strain 11.05~12.02%, the curve of the stress σx is again in a convex shape which is similar to the d- e-f stage. In the r-2-s-t stage of the strain range 12.02~13.07%, the stress σx of this stage drops down sharply to minimum point 2 and then rises up quickly to point t in Fig. 8(a), which corresponds that the dislocation annihilations simultaneously release the strain energy of the system.

Fig. 8(b) shows the σy-εy curve of the system under biaxial strain during the process of the deviatoric deformation. By comparing Fig. 8(a) and (b), we can see the stress σy and σy are just complementary, which may be resulted by the significant rigidity of the crystal system in response to an applied strain. Although the σx is much lower at same time, the failure and fracture of the sample does not occur under the deviatoric deformation. This indicates that σy in higher positive complementary state corresponding to lower state σx is helpful to suppress to nucleation of nanovoids or micro cracks. The atomic crystal of the PFC exhibits a significant rigidity in response to an applied strain. A more constructive way to model the collective movement of the dislocations of the atomic crystals would be accurately consider a phonon of wave term [83] in the dynamics. If such modes were considered, the collective motion of the dislocations in response to the applied strain would naturally be enhanced, and reversible movement [[87], [88], [89]] of homogenous nucleated dislocation can occur easily.

3.5. Nucleation and growth of grain for three different stages

3.5.1. Mechanism of nucleation and growth of deformed grain

For the sake of the simplicity to show the cooperative migration of the GB dislocations, the alternating arrangement of the composite vector Bi here indicates the alternating arrangement of the GB dislocations. Schematic diagram of the initial GB dislocation structure Bi=bi+bi+1 in the two grain system before the nucleation of the deformed grain is shown in Fig. 9(a) and (b) shows that the original GB with the dislocation Bi splits into two planar ATGBs under the deviatoric deformation in Fig. 5(d), and also gives the sketch map of the nucleation and generation of new grain 3 and 4 with orientation θ3 and θ4, respectively, where θ3 =θ4 = θ = 0°. In the area swept by the ATGB, the orientation of the original crystal is changed through local lattice rotation. This process is similar to that reported in the Ref. [31], i.e, the splitting transformation of the GB of nanocrystalline materials under the stress occurs.

Fig. 9.

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..


Three types of nucleation mode are observed by the PFC simulation. The first mode of the nucleation is that the GB splits into two ATGBs. When the strain reaches 0.0381, it induces the nucleation of two new grains with orientation 0° through the original GB splitting into two columns of the ATGBs as I and II, or III and IV, which is the first time of the grain nucleation, shown in Fig. 5(d). The gap areas between the ATGBs I and II, or III and IV, are just the two new grains with new orientation angle 0° (Such “gaps” between terminations of the dislocation arrays can be considered as disclination dipoles [31,34]). In this stage, the system transforms into four grain structures. The orientation angles of the four grains are respectively 4°, 0°, -4°, 0°. In the process of the migration of the ATGB, the two new deformed grains (DGs) with orientation angle 0° continuously grow and consume the original grains with orientation 4° or -4° and make them shrink, as shown in Fig. 5(d)-(i). This case may be have some similar to that of report in Ref. [31]. The second mode of the nucleation is owing to the reaction between the dislocations of different ATGBs. When the ATGB II meets III, or the ATGB I meets IV, the dislocation between different ATGB forms the dipole of the dislocation pair. At this time, the original grains vanish completely, and the whole region of the system transforms into a single crystal with 0° orientation without the GB structure, in which there are only the highest local dislocation density (each place is of four dislocations, in Fig. 6(d)-(e)) in a narrow distortion band [80] where the reaction of the dipole of the dislocation occurs, as shown in Figs. 5(j) and 6 (b)-(d). After that, with the strain continuously increasing, a strong torsion effect on the region of the dipole dislocation occurs to cause strong distortion of the softened lattice [27] in the area. In this process of the nucleation, the configuration change of the dipole of the dislocation pair Bi, and the generation and annihilation of the pair of the Frank sessile dislocations by climbing, and also the multiplication of the pair of purple color dislocation occur, respectively, as shown in Fig. 6(b)-(d). Thus it is just that the dislocation in different ATGB meets and reacts with each other and causes their configuration exchange [27,34] to nucleate the deformed grain with 0° orientation. We consider that the process of the formation and decomposition of the dipole dislocation pair, or to say the reaction process of the dislocation pair corresponds to the nucleation process and its microscopic mechanism is similar to that of the grain during dynamic recrystallization at high temperature [27,[88], [89], [90], [91], [92]]. With the separation of the dislocation pairs in the single crystal system, the second growth of two new grains with orientation 4° or -4° starts, and also four rows of the ATGB glide away again, as shown in Fig. 5(m). During the process of the second grain growth, the four-sub-grain structure occurs again in the system. With the increase of the applied strain, these new grains with orientation 4° or -4° grow by migration of the ATGB, and consume those two DGs with higher energy and orientation angle 0°. In the end, the ATGB of the two grains with orientation 4° or -4° meet and pass through each other, and then the DG is consumed up and vanishes. The whole system again restores the original bicrystal state with orientation 4° or -4°, as shown in Fig. 5(n).

The third mode of the nucleation is that two rows of the dislocation of different ATGBs are oncoming and pass through slipping and crossing without dislocation reaction. With the increase of the strain, the GB splits again into two columns of the ATGB shown in Fig. 5(n)-(q). Finally, the combination of the two grains makes the system become a single crystal structure with only one orientation 0°. In the present works, the GB dislocations gliding collectively in cooperation away from the original GB is different from that of the dislocation in a sessile or a glissile EGBD in Ref. [30], and also rather than the dislocation collectively sliding along the original GB under a shearing action, as shown in Ref. [12,75], in which the GB position does not change, and only the shape of the grain changes without any generation of new grain [12,19]. The results in present work are also very different from that of the sample A at low temperature [37], in which only the grain grows without any multiplication of the dislocations.

3.5.2. The rule of grain growth under the deviatoric deformation

For the growth of the grain in the three stages on the above statements, the curve of the changing area of the grain vs time is shown in Fig. 10. We consider that the grain growth in the first and third stages is of the similar process of the driven nucleation and growth of the grain, yet, the nucleation of the dynamic recrystallization grain in the second stage is owing to the multiplication and annihilation of the dipole dislocation pair Bi shown in Fig. 6(a)-(f). Usually the dislocation density of the GB in the deformed grain is higher than that inside the grain. Therefore, when the GB dislocation density increases under the deviatoric deformation to reach the critical value, ρc, the nucleation of the dynamic recrystallization grain is priority on the GB of the deformed grain [91]. The annihilation and multiplication of the dislocations in this nucleation process can be described by the improved Kocks-Mecking model [27] which can be used to show the process of the cycle dynamic recovery. The recrystallization grain consumes the DG with orientation 0° shown in Fig. 5(k)-(n). The processes of the three stages of the grain growth all belong to that of the growth under the deviatoric strain.

Fig. 10.

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).


We can write the grain growth formula by following the formula in Refs. [92,93] as below:

$A-{{A}_{0}}=\alpha \times {{t}^{\beta }}$

where the α and β are the fitting coefficients, A0 is the fixed parameter (the initial area), here A0 is set as 1.5. We obtain the formula of the grain growth for the three stages respectively by fitting the data in Fig. 10, which are given in Table 2. It can be seen in Table 2 that α and β coefficients of the formula are almost the same for the new grain with 0° orientation in the first and third stages. This expresses that the rule of the grain growth of the two stages is similar. However, the second stage is the growth of the dynamic recrystallization grain with orientation angle 4° or -4°. It can be seen from Table 2 that the time index of the three stages is approximately equal to 1.9. For convenience, the β is taken as integer 2 here. The grain growth in the three stages are approximately consistent with the rule t2, while only the growing coefficients α is different. The growing coefficient α=7.6×10-6 for the second stage is about as four times as that of the first and third stages. This indicates the growing speed of the recrystallization grain in area during the second stage is fast about as four times as α=1.67×10-6 of the DG in first and third stages. The reason is that the grain growth of the first and third stages is of the process of the accumulation of the strain energy of the system, during which the external strain acts to drive the grain boundary splitting, therefore, the strain energy inside the DG increases [66,77]; However, the grain growth of the second stage is of the process of the release of the strain energy by dislocation gliding. The area A of the DG can be gotten as A=Lx, where L is a constant equaling to the length of the sample. Then, the grain growth of the area formula (12) can be rewritten as

$x-{{x}_{0}}=\frac{\alpha }{L}\times {{t}^{1.9}}\approx \frac{\alpha }{L}\times {{t}^{2}}\text{,}$

Table 2   The fitting formula of grain growth of the PFC simulation in area.

Stages: Grain:First stage of growth: Deformed grain with 0° orientationSecond stage of growth: Recrystallization grain with 4° and -4° orientationThird stage of growth: Deformed grain with 0° orientation
The fitting formula:A-1.5=1.68×10-6×t1.89A-1.5=7.6×10-6×t1.85A-1.5=1.67×10-6×t1.91
fitting coefficient:α = 1.68 × 10-6, β = 1.89α = 7.6 × 10-6, β = 1.85α = 1.67 × 10-6, β = 1.91

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Unit of t : time step (ts); Unit of A: area (grid numbers).

which transforms from the area growth to the length growth for the grain, and then the speed of the growth can be gotten

${{v}_{\text{GB}}}=\frac{dx}{dt}\approx \frac{2\alpha }{L}t=\frac{2\alpha }{L}\cdot \frac{\varepsilon }{{\dot{\varepsilon }}}\text{.}$

where $\dot{\varepsilon }$ is a constant. It can be seen that the speed of the growth linearly increase with the strain. In literature [34,[92], [93], [94]], the grain growth is simulated and analyzed under the strain, and the results [14,76] of the grain growth accelerated by the strain are also obtained and are close to the law of the time square t2. From the formula of the diameter of the grain growth given in Ref. [90,91,96], we can see the time index β of the grain growth in formula (12) without external applied stress or strain (here refer to the curvature-driven growth of grain) is about 1/2 [96]. Under the strain of the deviatoric deformation, the rate of grain growth for the three stages is obvious faster than that of the curvature-driven growth [[97], [98], [99], [100], [101], [102]] of the grain. Here the GB dislocation sliding synchronously and collectively is also different from that of the report in Ref. [6,9], in which the dislocation slides under the shear strain only along the GB and no new grain generation. In contract, the original GB in this work splits into two ATGBs or the reaction of the dislocation pair generates new sub-grain with new orientation and grows through the ATGB dislocation gliding to release the strain energy.

3.5.3. Dynamic of the deformed grain growth

Mcreynods et al. [28] have pointed out that the translation of the lattice can be fully compensated by the dislocation climb. Following to Ref. [5,6] for the case of no sliding of the small angle GB, the tangential velocity vt of the migration of the GB is linear to the normal velocity vn of the GB. The main sources for grain rotation and the GB migration are stemmed from: 1) net torque and surface tension; 2) external stresses. Usually, the speed of the growth (dD/dt) is proportional to the driving force P [95,102] and a mobility M. Here, the GB migration is vertical to the GB direction during the deformed grain growth (DGG), while the slipping direction of the GB is along the GB. The dislocation Bi of the GB moves roughly along the close-packed direction [67] of the Burgers vector. Following to the GB coupling movement in Ref. [9,10], the dynamic equation can be obtained as

${{v}_{n}}=\frac{d{{D}_{x}}}{dt}={{M}_{ix}}{{P}_{ix}}+{{M}_{ex}}{{P}_{ex}}+{{\beta }_{x}}\frac{d{{D}_{y}}}{dt}$
${{v}_{\tau }}=\frac{d{{D}_{y}}}{dt}={{\text{S}}_{iy}}{{P}_{iy}}+{{S}_{ey}}{{P}_{ey}}+{{M}_{iy}}{{P}_{ix}}+{{M}_{ey}}{{P}_{ex}}+{{\beta }_{y}}\frac{d{{D}_{x}}}{dt}$

where t is the time. The migration distance of the ATGB is denoted as Dx and the distance along the ATGB is denoted as Dy. Mi and Me are the coefficient of ATGB migration, respectively. S and β are respective the coefficient of the sliding or coupling [9,10]. Pi and Pe are the driving force by internal stress or external stresses, respectively. For the pure STGB, dislocations do not slip along the GB. Then Six = 0 and Sey = 0. In this PFC simulation, the grain here is the columnar grain and the ATGB of the grain is planar. Hence, there is no the driving force by the curvature. No difference of the dislocation density is between two sides of the ATGB. Hence the effect of the internal stress can be ignored. We only consider the migration driven by the external stress. Then, the equation gotten from Eqs. (15a) and (15b) is as following

${{v}_{n}}={{M}_{ex}}{{P}_{ex}}+{{\beta }_{x}}{{v}_{\tau }}$
${{v}_{\tau }}={{M}_{ey}}{{P}_{ey}}+{{\beta }_{y}}{{v}_{n}}$

According to Ref. [9], we have βxy=πθ/180, and ${{\beta }_{x}}\cdot {{\beta }_{y}}$ for the small angle STGB under θ (in degrees) < 8° can be ignored owing to a second order small amount of that. Then, the dynamic equations under the deformation can be gotten from Eqs. (16a) and (16b), as

${{v}_{n}}=\frac{{{M}_{ex}}{{P}_{ex}}+{{\beta }_{x}}{{M}_{ey}}{{P}_{ey}}}{1-{{\beta }_{x}}{{\beta }_{y}}}\approx {{M}_{ex}}{{P}_{ex}}+{{\beta }_{x}}{{M}_{ey}}{{P}_{ey}}=(1+{{\beta }_{x}}){{M}_{ex}}{{P}_{ex}}$
${{v}_{\tau }}=\frac{{{M}_{ey}}{{P}_{ey}}+{{\beta }_{y}}{{M}_{ex}}{{P}_{ex}}}{1-{{\beta }_{x}}{{\beta }_{y}}}\approx {{M}_{ey}}{{P}_{ey}}+{{\beta }_{y}}{{M}_{ex}}{{P}_{ex}}=(1+{{\beta }_{y}}){{M}_{ey}}{{P}_{ey}}$

where we assume Mex=Mey and Pex=Pey for simplicity. The driving force [95] on the GB by the work input into the system can be written as the form of the linear rate of mechanical work at any moment during deformation, for example

${{P}_{ex}}\propto \eta \dot{\varepsilon }{{\sigma }_{x}}\text{and}{{P}_{ey}}\propto \eta \dot{\varepsilon }{{\sigma }_{y}}$

where σx and σy is the external applied stress on the sample surface during the deformation, and η is a proportional constant and can be approximated as η = 1. The GB mobility Mex can be given in an Arrhenius-type equation

${{M}_{ex}}={{M}_{0}}\exp (\frac{{{Q}_{GB}}}{RT})$

where Mo a constant, and R is the gas constant, and for isothermal grain growth, QGB is the activation energy. The grain growth speed along x direction is

$\begin{array}{*{35}{l}}{{v}_{n}}={{v}_{GB}}=\frac{d{{D}_{x}}}{dt}=\eta (1+{{\beta }_{x}}){{M}_{ex}}{{P}_{ex}}=\eta \dot{\varepsilon }(1+{{\beta }_{x}}){{M}_{ex}}{{\sigma }_{x}} \\ =\eta (1+{{\beta }_{x}}){{M}_{ex}}\cdot \frac{E}{1+\nu }\dot{\varepsilon }\varepsilon =\eta (1+{{\beta }_{x}}){{M}_{ex}}\cdot {{(\dot{\varepsilon })}^{2}}\frac{E}{1+\nu }t \\ \end{array}$

where $\varepsilon ={{\dot{\varepsilon }}_{x}}t$. Then the $\frac{d{{D}_{x}}}{dt}$ is linear increasing with the strain in formula (20) and agreement with the formula (14) and the result of Ref. [83]. Then the migration distance of the GB of the deformed grain is expressed as

$\begin{array}{*{35}{l}} {{D}_{x}}=\frac{1}{2}(1+{{\beta }_{x}}){{M}_{ex}}\frac{\eta E}{1+\nu }{{(\dot{\varepsilon })}^{2}}\cdot {{t}^{2}} \\ =\frac{1}{2}(1+{{\beta }_{x}}){{M}_{ex}}\frac{\eta E}{1+\nu }{{(\varepsilon )}^{2}}\propto {{(\varepsilon )}^{2}} \\ \end{array}$

The regularity of the DGG given by Eq. (21) is good agreement with Eq. (13). Comparing the Eqs. (14) and (20), we can get

$\alpha =(1+{{\beta }_{x}}){{\text{M}}_{e}}\cdot \frac{\eta E\text{ }\!\!{}^\circ\!\!\text{ }L}{2(1+\nu )}\cdot {{(\dot{\varepsilon })}^{2}}$

It can be seen that the coefficient α is related to the E, L, Me, ν, and ${{\dot{\varepsilon }}_{x}}$. The GB mobility is about M0 = 2.5*10−6(m4/Js) for pure Cu metal [68,76], and Possion’s ratio ν = 0.308, ε = 7.1%, η = 1, ${{\beta }_{x}}=\sin (\frac{\theta }{2})=$0.06, ${{\dot{\varepsilon }}_{x}}$=7.2 × 10−6, T = 750 K corresponding to r = 0.30 temperature parameter [27] of the PFC simulation. Then, the migration speed of the GB of Cu metal can be gotten as ${{V}_{GB}}\approx $ 3.1 (cm/s) by the formula (20), which is good agreement to the result 3.0 (cm/s) of the experiment [[100], [101], [102]].

4. Conclusions

Although the simple bicrystal system is used as the research sample in the present paper, this simplified system is very practical for studying the cooperative dislocation motion of the GB migration under the action of biaxial strain at high temperature. It is possible to observe the configuration change, proliferation and annihilation of the dislocations by using the PFC simulation and the continuum modeling. At the same time, the different growth mechanisms of new grain and the change in grain orientation and the phenomenon of cross-slipping and traversing of the GB dislocations are also observed. The main conclusions are as below:

1)Three types of nucleation modes are observed by the PFC simulation: The first mode of the nucleation is generated by the GB splitting into two ATGBs; The second mode is of the reaction of the generation and annihilation of a pair of Frank sessile dislocation in 2D, which is similar to the nucleation of the grain in the dynamic recrystallization; The third mode is that two rows of the dislocations of these ATGBs are oncoming and pass by crossing each other. The first mode and third mode are similar in general.

2)The cooperative dislocation motion of the GB under the deviatoric deformation accompanies with local plastic flow and with the propagation of the localized strain energy. When the system is in the process of the recrystallized grain growth, the system energy is in an unstable state due to the strain energy releasing quickly, which causes that reverse movement of the plastic flow occurs and involves local nonlinear physical mechanism.

3)Due to the nucleation in different modes, the mechanism of the grain growth by means of the ATGB 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 curvature-driven grain growth without external driving force. The area growth of the process is approximately proportional to the time square under the deviatoric deformation. The growth rate of the grain area of the recrystallization for the second nucleation mode is about as four times as that of the deformed grain for the nucleation of the first and third modes.

Acknowledgments

This work is supported by National Nature Science Foundation of China (Nos. 51161003 and 51561031); Nature Science Foundation of Guangxi Province (No. 2018GXNSFAA138150).

Appendix A. The relationship between vector {hi}, {bi} and {Bi}

Fig. 4(a) shows the directions of the set of vectors {hi} for a half of excess atomic plane of the edge dislocation, where hi=hi+6. The direction vectors of hi are given according to Fig. 4(a) as below:

${{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{1}}=[\frac{\sqrt{3}}{2},\frac{1}{2}]{{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{2}}=[\frac{\sqrt{3}}{2},-\frac{1}{2}]{{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{3}}=[0,-1]{{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{4}}=[-\frac{\sqrt{3}}{2},-\frac{1}{2}]{{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{5}}=[-\frac{\sqrt{3}}{2},\frac{1}{2}]{{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{6}}=[0,1]$

Fig. 4(b) shows the directions of the set of vectors {bi} which correspond to the Burgers vector of the half of excess atomic plane of the edge dislocation and have bi=bi+6. The direction of bi are given according to Fig. 4(b) as below:

${{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{1}}=[-\frac{1}{2},\frac{\sqrt{3}}{2}],{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{2}}=[\frac{1}{2},\frac{\sqrt{3}}{2}],{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{3}}=[1,0],{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{4}}=[\frac{1}{2},-\frac{\sqrt{3}}{2}],{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{5}}=[-\frac{1}{2},-\frac{\sqrt{3}}{2}],{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{6}}=[-1,0]$

The set of vectors {hi} and the set of vector {bi} satisfy the relationship ${{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{i}}\bot {{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{i}}$, where “⊥” indicates vertical. All six Burgers vectors {bi} can be expressed by the basic b1 and b2 in combination. For example,

${{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{3}}={{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{2}}-{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{1}};{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{4}}=-{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{1}};{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{5}}=-{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{2}};{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{6}}={{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{1}}-{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{2}}.$

Here bi can be regarded as a partial dislocation which cannot exist alone, but it only exists in pairing. We define the composite vector Bi of the dislocation pair, ${{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{i}}={{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{i}}+{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{i+1}}$, where ${{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{i}}={{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{i+6}}$, $i=1,2,3,4,5,6$. The Bi of the dislocation pair can be regarded as a full dislocation. Fig. 4(c) shows the directions of a set of composite vectors {Bi}. The six possible composite vectors Bi constituted by Bi=bi+bi+1 are shown in Fig. 4(c). These sets of {Bi} and {bi} and {hi} constitute a complete set, respectively.

The {Bi} can be expressed by two basic vectors, b1 and b2, and also be expressed by two basic vectors B1 and B2. The details of the {Bi} can be written by b1 and b2, or B1 and B2 as below

${{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{1}}={{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{1}}+{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{2}}=\sqrt{3}[0,1]=-{{\vec{B}}_{4}}$

${\overset{\scriptscriptstyle\rightharpoonup}{B}}_{2}={\overset{\scriptscriptstyle\rightharpoonup}{b}}_{2}+{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{3}=2{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{2}-{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{1}=\frac{\sqrt{3}}{2}[\sqrt{3},1]=-{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{5}$

${\overset{\scriptscriptstyle\rightharpoonup}{B}}_{3}={\overset{\scriptscriptstyle\rightharpoonup}{b}}_{3}+{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{4}={\overset{\scriptscriptstyle\rightharpoonup}{b}}_{2}-2{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{1}=\frac{\sqrt{3}}{2}[\sqrt{3},-1]=-{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{6}={\overset{\scriptscriptstyle\rightharpoonup}{B}}_{2}-{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{1}$

${{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{4}}={{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{4}}+{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{5}}=-({\overset{\scriptscriptstyle\rightharpoonup}{b}}_{1}+{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{2})=\sqrt{3}[0,1]=-{{\vec{B}}_{1}}$

${\overset{\scriptscriptstyle\rightharpoonup}{B}}_{5}={\overset{\scriptscriptstyle\rightharpoonup}{b}}_{5}+{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{6}=-(2{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{2}-{\overset{\scriptscriptstyle\rightharpoonup}{b}}=\frac{\sqrt{3}}{2}[\sqrt{3},1]=-{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{2}$

${\overset{\scriptscriptstyle\rightharpoonup}{B}}_{6}={\overset{\scriptscriptstyle\rightharpoonup}{b}}_{6}+{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{1}=-{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{2}+2{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{1}=\frac{\sqrt{3}}{2}[\sqrt{3},-1]=-{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{3}=-({\overset{\scriptscriptstyle\rightharpoonup}{B}}_{1}-{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{2})$

It can see that the direction of the sets of {Bi+1} and {hi} are identical, and the {bi} can also be expressed by two composite vectors, B1 and B2

${{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{1}}=\frac{1}{3}(2{{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{1}}-{{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{2}}),{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{2}}=\frac{1}{3}({{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{2}}-{{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{1}}),{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{3}}=\frac{1}{3}(2{{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{2}}-3{{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{1}}),$

${{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{\text{4}}}=\frac{1}{3}({{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{2}}-2{{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{1}}),{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{\text{5}}}=\frac{1}{3}({{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{1}}-{{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{2}}),{{\overset{\scriptscriptstyle\rightharpoonup}{b}}_{\text{6}}}=\frac{1}{3}(3{{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{1}}-2{{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{2}}),$

Comparing Fig. 4(a) with Fig. 4(c), then we can find the direction of {Bi} and {hi} have ${{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{1}}\parallel \sqrt{3}{{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{6}}$, ${{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{2}}\parallel \sqrt{3}{{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{1}}$, ${{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{3}}\parallel \sqrt{3}{{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{2}}$, ${{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{4}}\parallel \sqrt{3}{{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{3}}$, ${{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{5}}\parallel \sqrt{3}{{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{4}}$, ${{\overset{\scriptscriptstyle\rightharpoonup}{B}}_{6}}\parallel \sqrt{3}{{\overset{\scriptscriptstyle\rightharpoonup}{h}}_{5}}$, where “$\parallel $” indicates parallel. As can be seen from Fig. 4(a)-(c), the vector directions of the set {Bi} are reversed by 30° with respect to the vector directions of the set {bi} correspondingly, and is rotated forward by 60° with respect to the vector directions of the set {hi} correspondingly (Fig. A1, Fig. A2).

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.


Abbreviations: GB, grain boundary; GBD, grain boundary dislocation; EGBD,extrinsic grain boundary dislocation; MD, molecular-dynamic; TPF, Traditionalphase field; PFC, Phase field crystal; CDM, coordinative dislocation movement;STGBs, Symmetric tilt grain boundaries; ATGB, Asymmetric tilt grain boundaries;DG, deformed grain; OG, original grain; DGG, deformed grain growth.

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AbstractMolecular dynamics simulations were used to study the effect of applied force and grain boundary misorientation on grain boundary sliding in aluminum at 750 K. Two grains were oriented with their 〈1 1 0〉 axes parallel to their boundary plane and one grain was rotated around its 〈1 1 0〉 axis to various misorientation angles. For any given misorientation, increasing the applied force leads to three sliding behaviors: no sliding, constant velocity sliding and a parabolic sliding over time. The last behavior is associated with disordering of atoms along the grain boundary. For the second sliding behavior, the constant sliding velocity varied linearly with the applied stress. A linear fit of this relationship did not intersect the stress axis at the origin, implying that a threshold stress for sliding exists. This threshold stress was found to decrease with increasing grain boundary energy. The ramifications of this finding for modeling grain boundary sliding in polycrystals are discussed.]]>

B. Liu, K. Wang, F. Liu, Y. Zhou, J. Mater. Sci. Technol. 34 (2018) 1359-1369.

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AbstractWe apply the semi-grand-canonical Monte Carlo method with an embedded-atom potential to study grain boundary (GB) premelting in Cu-rich Cu–Ag alloys. The Σ5 GB chosen for this study becomes increasingly disordered near the solidus line while its local chemical composition approaches the liquidus composition at the same temperature. This behavior indicates the formation of a thin layer of the liquid phase in the GB when the grain composition approaches the solidus. The thickness of the liquid layer remains finite and the GB can be overheated/oversaturated to metastable states slightly above the solidus. The premelting behavior found by the simulations is qualitatively consistent with the phase-field model of the same binary system presented in Part I of this work [Mishin Y, Boettinger WJ, Warren JA, McFadden GB. Acta Mater, in press]. Although this agreement is encouraging, we discuss several problems arising when atomistic simulations are compared with phase-field modeling.]]>

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AbstractMolecular dynamics (MD) simulations are used to study diffusion-accommodated creep deformation in nanocrystalline molybdenum, a body-centered cubic metal. In our simulations, the microstructures are subjected to constant-stress loading at levels below the dislocation nucleation threshold and at high temperatures (i.e., T > 0.75Tmelt), thereby ensuring that the overall deformation is indeed attributable to atomic self-diffusion. The initial microstructures were designed to consist of hexagonally shaped columnar grains bounded by high-energy asymmetric tilt grain boundaries (GBs). Remarkably the creep rates, which exhibit a double-exponential dependence on temperature and a double power-law dependence on grain size, indicate that both GB diffusion in the form of Coble creep and lattice diffusion in the form of Nabarro–Herring creep contribute to the overall deformation. For the first time in an MD simulation, we observe the formation and emission of vacancies from high-angle GBs into the grain interiors, thus enabling bulk diffusion.]]>

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Y. Mishin, W.J. Boettinger, J.A. Warren G.B. McFadden, Acta Mater. 57 (2009) 3771-3779.

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AbstractThe rich nature of the premelting transition of grain boundaries in solid solutions is analyzed. Part I of this paper uses a multi-phase field model, whereas Part II employs atomistic Monte Carlo simulations. To enable comparison, Cu-rich Cu–Ag solid solutions are chosen for study. In the phase-field model, a system composed of two grains and a liquid phase is treated with three phase field parameters and with a realistic bulk thermodynamic description of Cu–Ag alloys obtained with the CALPHAD approach. Several different computation methods are employed, both rigorous and approximate, to examine the premelting behavior and relate it to the so-called “disjoining potential” between the solid–liquid interfaces in the grain boundary region. Depending on the grain boundary energy, temperature and grain composition chosen, several different classes of premelting transitions have been detected. As the grain concentration approaches the solidus line, one class shows a premelted layer whose thickness diverges continuously to infinity (complete wetting). Another class shows a discontinuity of the premelted layer thickness, exhibiting a first-order thin-to-thick transition prior to continuous thickening to infinity at the solidus line. In other cases, a metastable grain boundary state can exist above the solidus line, indicating the possibility of superheating/supersatuation of the grains together with the grain boundary. The possibility of such transitions has been predicted previously for generic thermodynamics by many authors. The results of the current investigation are compared with the atomistic calculations for the Cu–Ag system in Part II of this work.]]>

K.R. Elder, M. Katakowski, M. Haataja, Phys. Rev. Lett. 88 (2002), 245701.

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A new model of crystal growth is presented that describes the phenomena on atomic length and diffusive time scales. The former incorporates elastic and plastic deformation in a natural manner, and the latter enables access to time scales much larger than conventional atomic methods. The model is shown to be consistent with the predictions of Read and Shockley for grain boundary energy, and Matthews and Blakeslee for misfit dislocations in epitaxial growth.

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Bulk-immiscible binary systems often form stress-induced miscible alloy phases when deposited on a substrate. Both alloying and surface dislocation formation lead to the decrease of the elastic strain energy, and the competition between these two strain-relaxation mechanisms gives rise to the emergence of pseudomorphic compositional nanoscale domains, often coexisting with a partially coherent single phase. In this work, we develop a phase-field crystal model for compositional patterning in monolayer aggregates of binary metallic systems. We first demonstrate that the model naturally incorporates the competition between alloying and misfit dislocations, and quantify the effects of misfit and line tension on equilibrium domain size. Then, we quantitatively relate the parameters of the phase-field crystal model to a specific system, CoAg/Ru(0001), and demonstrate that the simulations capture experimentally observed morphologies.

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The world of two-dimensional crystals is of great significance for the design and study of structural and functional materials with novel properties. Here we examine the mechanisms governing the formation and dynamics of these crystalline or polycrystalline states and their elastic and plastic properties by constructing a generic multimode phase field crystal model. Our results demonstrate that a system with three competing length scales can order into all five Bravais lattices, and other more complex structures including honeycomb, kagome, and other hybrid phases. In addition, nonequilibrium phase transitions are examined to illustrate the complex phase behavior described by the model. This model provides a systematic path to predict the influence of lattice symmetry on both the structure and dynamics of crystalline and defected systems.

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We investigate elastic energy-driven grain boundary migration in a strained copper bicrystal using an atomistically informed phase field model. In a bicrystal experiencing a uniform strain, the softer grain has a lower energy density and grows at the expense of the harder grain. In a bicrystal experiencing heterogeneous strain, the softer grain has a higher energy density, yet it still grows. Our findings suggest that the softer grain will grow, irrespective of the difference in the energy densities.

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In crystalline materials, plastic deformation occurs by the motion of dislocations, and the regions between individual crystallites, called grain boundaries, act as obstacles to dislocation motion. Grain boundaries are widely envisaged to be mechanically static structures, but this report outlines an experimental investigation of stress-driven grain boundary migration manifested as grain growth in nanocrystalline aluminum thin films. Specimens fabricated with specially designed stress and strain concentrators are used to uncover the relative importance of these parameters on grain growth. In contrast to traditional descriptions of grain boundaries as stationary obstacles to dislocation-based plasticity, the results of this study indicate that shear stresses drive grain boundaries to move in a manner consistent with recent molecular dynamics simulations and theoretical predictions of coupled grain boundary migration.

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AbstractThe mechanical behavior and the kinetics of shear banding in bulk metallic glasses was investigated at room and liquid nitrogen temperature using the acoustic emission (AE) technique. It was demonstrated that the intensive AE reflecting the activity of strongly localized shear bands at room temperature vanishes at the transition of plastic flow from serrated to non-serrated with a reduction in temperature. The disappearance of AE clearly suggests that the shear band propagation velocity significantly decreases at low temperature, and sliding along the principle shear band is observed at the machine-driven rate.]]>

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The critical dynamics of dislocation avalanches in plastic flow is examined using a phase field crystal model. In the model, dislocations are naturally created, without any ad hoc creation rules, by applying a shearing force to the perfectly periodic ground state. These dislocations diffuse, interact and annihilate with one another, forming avalanche events. By data collapsing the event energy probability density function for different shearing rates, a connection to interface depinning dynamics is confirmed. The relevant critical exponents agree with mean field theory predictions.

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We demonstrate reversible movement of 1/2[11[over ]0](110) dislocation loops generated from nanodisturbances in a beta-titanium alloy. High resolution transmission electron microscope observations during an in situ tensile test found three reversible deformation mechanisms, nanodisturbances, dislocation loops and martensitic transformation, that are triggered in turn with increasing applied stress. All three mechanisms contribute to the nonlinear elasticity of the alloy. The experiments also revealed the evolution of the dislocation loops to disclination dipoles that cause severe local lattice rotations.

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AbstractA theoretical model is suggested which describes cooperative action of grain boundary (GB) sliding and rotational deformation in mechanically loaded nanocrystalline materials. Focuses are placed on the crossover from GB sliding to rotational deformation occurring at triple junctions of GBs. In the framework of the model, gliding GB dislocations at triple junctions of GBs split into dislocations that climb along the adjacent boundaries. The splitting processes repeatedly occurring at triple junctions give rise to climb of GB dislocation walls that carry rotational deformation accompanied by crystal lattice rotation in grains of nanocrystalline materials. The role of GB sliding, rotational deformation and conventional dislocation slip in high-strain-rate superplastic flow in nanocrystalline materials is discussed.]]>

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The quest for multiferroic materials with ferroelectric and ferromagnetic properties at room temperature continues to be fuelled by the promise of novel devices. Moreover, being able to tune the electrical polarization and the paramagnetic-to-ferromagnetic transition temperature constitutes another current research direction of fundamental and technological importance. Here we report on the first-principles-based prediction of a specific class of materials--namely, R2NiMnO6/La2NiMnO6 superlattices where R is a rare-earth ion--that exhibit an electrical polarization and strong ferromagnetic order near room temperature, and whose electrical and ferromagnetic properties can be tuned by means of chemical pressure and/or epitaxial strain. Analysis of the first-principles results naturally explains the origins of these highly desired features.

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AbstractWe have made measurements of the temporal and spatial features of the evolution of strain during the serrated flow of Pd40Ni40P20 bulk metallic glass tested under quasistatic, room temperature, uniaxial compression. Strain and load data were acquired at rates of up to 400 kHz using strain gages affixed to all four sides of the specimen and a piezoelectric load cell located near the specimen. Calculation of the displacement rate requires an assumption about the nature of the shear displacement. If one assumes that the entire shear plane displaces simultaneously, the displacement rate is approximately 0.002 m s–1. If instead one assumes that the displacement occurs as a localized propagating front, the velocity of the front is approximately 2.8 m s−1. In either case, the velocity is orders of magnitude less than the shear wave speed (∼2000 m s−1). The significance of these measurements for estimates of heating in shear bands is discussed.]]>