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

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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.

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.

Section snippets

PFC model

The free energy functional F˜of the system in PFC model given in Refs. [46,47] is asF˜=f(ρ(x(1+ε),y(1ε)))dV=[f(ρ(x,y))+Eext(ε,x,y)]dV

where the local free energyf(ρ) is asf(ρ(x,y))=12eρ2+14ρ4+12ρ(1+2)2ρwhere ρis an order parameter corresponding to atomic density and e is a temperature parameter, and 2 the Laplace operator. Eextis the changed energy by external force, and is written asEext(ε,x,y)=Vextρwhere Vext is external force, and the detailed expression can be seen in Ref. [26]. The

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

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

Acknowledgements

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

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