Low-energy, Mobile Grain Boundaries in Magnesium

The strong basal texture that is commonly developed during the rolling of magnesium alloy and can even increase during annealing motivates atomic-level study of dislocation structures of both <0001> tilt and twist grain boundaries (GBs) in Magnesium. Both symmetrical tilt and twist GBs over the entire range of rotation angles θ between 0° and 60° are found to have an ordered atomic structure and can be described with grain boundary dislocation models. In particular, 30° tilt and twist GBs are corresponding to energy minima. The 30° tilt GB is characterized with an array of Shockley partial dislocations bp:-bp on every basal plane and the 30° twist GB is characterized with a stacking faulted structure. More interesting, molecular dynamics simulations explored that both 30° tilt and twist GBs are highly mobile associated with collective glide of Shockley partial dislocations. This could be responsible for the formation of the strong basal texture and a significant number of 30° misorientation GBs in Mg alloy during grain growth.

Scientific RepoRts | 6:21393 | DOI: 10.1038/srep21393 crystal. Between 0° and 60°, there is only one minimum energy GB at the rotation angle 30°. This is consistent with Electron Back Scatter Diffraction (EBSD) observations that a significant number of boundaries with 30° misorientation about the < 0001> direction is characterized in Mg alloy AZ31B during grain growth 11 ; and also satisfies the energetic criterion: low energy GBs favorably grow during annealing. The following issue is to test whether these low energy GBs are highly mobile.
symmetrical twist GB is semi-coherent and contains three sets of Shockley partial dislocations, as schematically shown in Fig. 2a. These dislocations have Burgers vectors of 1/3[0110], 1/3 [1010], and 1/3[1100] and separate the interface into two types of coherent structures: normal hexagonal close packed (HCP) structure (… ABABAB…) and stacking faulted (SF) structure (…ABABACAC…). A, B, and C represent three hexagonal close packed atomic planes with respect to (111) plane stacking in a face centered cubic (FCC) structure 19,20 . Due to the greater excess energy of the faulted structure than the normal HCP structure, misfit dislocation lines curve towards the SF regions associated with reducing the area of the faulted structure, decreasing interface excess energy as shown in Fig. 2b 19 . With increasing the rotation angle from 0° to 30° or decreasing the rotation angle from 60° to 30°, the density of misfit dislocations increases. In other words, the spacing between misfit dislocations decreases in association with the increase of the number of misfit dislocation lines, as discussed in FCC-FCC (111) interfaces [21][22][23] . The 30° (0001)-GB is a stacking faulted coherent interface with the stacking … ABABACACA…, where the atomic plane A is shared by the two crystals (Fig. 2c). The migration of the 30° (0001)-GB can be accomplished through the glide of Shockley partial dislocations with two possible mechanisms. Figure 2d shows a two-layer migration mechanism via the glide of a pair of Shockley partial dipole on two neighboring {0001} planes. Associated with the GB migration through the 2-layer mechanism, there is no macro-scale strain because of the net zero Burgers vector 24 . Figure 2e shows a single layer migration mechanism via successive gliding of single Shockley partial dislocation on every {0001} plane, resulting in a macro-scale shear strain. Such Shockley partial dislocations can be generated by either the interaction of non-basal dislocations with GBs or the nucleation of partial dislocation at GBs [25][26][27] . Thus, the 30° (0001) GB is thermodynamically preferred during grain growth because of the low excess energy and high mobility associated with the glide of Shockley partial dislocations 28 .
< 0001> symmetrical tilt GBs can be described with grain boundary dislocations as a tilt wall (atomic structures of several < 0001> -GBs are shown in Figure S2 in Supplementary) 29,30 . The atomic structure of the lowest energy 30° < 0001> -GB is shown in Fig. 3a,b. The right crystal has the stacks …ABABAB… and the left crystal has the stacks …ACACAC…. The migration of the GB towards the right involves the transformation of atomic planes B into atomic planes C. This can be accomplished with the glide of Shockley partial dipole b p above and −b p under the atomic plane B 31,32 . Therefore, the 30° < 0001> -GB can be represented as an array of Shockley partial dislocations on every {0001} plane with a repeatable sequence − b b : p p , as shown in Fig. 3c; and can be created by gliding a set of partial dislocation dipoles in a single crystal (Fig. 3d-f) 24 . Accompanying with the glide of one pair of partial dislocation dipole on the neighboring {0001} planes, one atomic plane B shifts into plane C. Such gliding of a pair dislocation dipole will not generate strains due to the net zero Burgers vector. This is similar to Σ 3{112} incoherent twin boundary in face centered cubic structure where three Shockley partial dislocations collectively glide on three (111) planes with the net zero Burgers vector 24,28,31,32 .
Owing to high mobility of Shockley partial dislocation on (0001), the 30° < 0001> -GB could be highly mobile corresponding to its dislocation structure. However, mechanical loading may not facilitate such motion because the net Peach-Kohler force acting on the partial dipole is equal to zero 33,34 . A generalized force, such as an elastic energy difference between the neighboring grains 18 , defect density difference in the neighboring grain 35 , or decreasing the curvature and/or area of grain boundaries 36 , can drive the migration of GBs. To test the mobility of the 30° misorientation GBs, we create a bi-crystal structure: a hexagonal pore grain is embedded in the matrix according to the misorientation of the 30° < 0001> -GB (Fig. 4a). The hexagonal pore grain has the side length of 6 nm. Six GBs have boundary planes {1120}. According to the dislocation structure of the 30° < 0001> -GB, grain boundaries in the bi-crystal can be characterized with three Shockley partial dislocation loops, b 1 , b 2 and b 3 , and their junctions, J 12 , J 23 and J 31 , as shown in Fig. 4b. The bi-crystal structure is relaxed under zero applied stresses at room temperature of 300 K. The details can be found in Figure S3 in Supplementary.
MD simulations demonstrated the migration of the GBs (also see Movie I in Supplementary) and revealed migration mechanisms of the GBs -the collective glide of Shockley partial dislocations -as evidenced by six snapshots in Fig. 5. To identify Burgers vectors of these Shockley partial dislocations that present in the GBs, we performed disregistry analysis across {0001} planes (Fig. 5g-i)). The results indicate that the six GBs can be represented as three sets of repeatable dislocation loops with Burgers vectors b 1 , b 2 , and b 3 on every two atomic planes and three junctions. The junction formed by two dislocation loops is a jog with Burgers vector of 1/3< 2110> (e.g., the junction formed by the dislocation loops b 3 and b 1 is referred to as the jog J 31 . Its Burgers vector is equal to b 3 − b 1 ). These jogs are mobile because their Burgers vectors are along the compact direction on {0110} planes. Thus, the hexagonal pore grain is surrounded by three mobile jogs, J 12 J 23 , and J 31 , and three Shockley partial dislocation loops b 1 , b 2 and b 3 , as shown in Fig. 4b. During MD relaxation, it is noticed that the curved segment of each dislocation loop commences to move, driven by reducing the curvature of the GBs. The two partials on the adjacent glide planes glide together because they attract each other due to the same Burgers vector while the opposite line sense as shown in Fig. 4b (or they form a pair of dislocation dipole as shown in Fig. 3c). Reducing the area of GBs in the bi-crystal motivates the continuous glide of partial dislocations and jogs. The three jogs and dislocations move and finally meet together. The summation of Burgers vectors of these dislocations is equal to zero. Thus, the hexagonal pore grain finally transforms into the same orientation as the surrounding grain, forming a single crystal (Fig. 5f).

Discussion
The strong basal texture is commonly developed during the rolling of Mg alloy (AZ31) and can even increase during annealing. Using Electron Back Scatter Diffraction (EBSD) analysis, a significant number of boundaries with 30° misorientation about the < 0001> direction is characterized in Mg alloy AZ31B during grain growth. From thermodynamical viewpoint, texture evolution is controlled by boundary anisotropy in energy and mobility. This Atomistic simulations so far revealed that both 30° < 0001> -GB and (0001)-GB are energetically favored during grain growth due to the low excess formation energy, compared with other < 0001> -GBs and (0001)-GBs, and even compared with < 1120> and < 0110> tilt grain boundaries ( Figure S5 in Supplementary) 29,30 . The 30° (0001)-GB is characterized to be a stacking faulted structure and can migrate through the glide of Shockley partial dislocation. Atomic structure of the 30° < 0001> -GB is characterized to contain a repeatable sequence of Shockley partial dislocation dipoles … − b b : p p …. Such dislocation patterns imply that 30° < 0001> -GBs are highly mobile due to the easy glide of Shockley partial dislocation on (0001) plane. Using molecular dynamics simulation in a bi-crystal, we demonstrated the easy migration of 30° < 0001> -GBs through collective glide of Shockley partial dislocation dipoles and their junctions. It is noticed that the migration of these GBs does not result in macro-scale shear strains because of the net zero Burgers vector. The similar phenomena and mechanisms have been observed and demonstrated in nanotwinned Cu and Ag, and phase transformation in InAs nanowirs using in situ microscopes and atomistic simulations [31][32][33] , wherein three sets of Shockley partial dislocations with the net zero Burgers vector collectively glide, resulting in zero-strain twinning and detwinning. Migration of such GBs does not require mechanical loading due to the net zero PK force, but can be driven by decreasing grain boundary curvature, or by a generalized force, such as an elastic energy difference and/or the defect density difference between the neighboring grains, which is generally resulted after several mechanical deformation.
However, the conclusion drawn so far is based on molecular statics/dynamics calculations for ideal 30° < 0001> -GBs and 30° (0001)-GBs in Mg, regardless of temperature, impurity, and additional grain boundary defects including vacancy, interstitial, and disconnections etc. In real materials, such defects are often present within GBs. In addition, materials were synthesized or treated at different temperatures and/or during different mechanical processes, internal stresses during these processes will be built and may change the structure of GBs. Even for most thermally stable twin boundaries in fcc metals, GBs associated with twin orientation relationships can contain different facets that vary with temperature and impurity, which corresponds to the faceting-roughening transformation mechanisms 37 . Corresponding to the dislocation structure of ideal 30° < 0001> -GBs and 30° (0001)-GBs in Mg, high temperature will reduce the Pierles stress of Shockley partial dislocations, facilitating the motion of Shockley partial dislocations. As a consequence, GBs deviating from ideal 30° < 0001> -GBs will be thermodynamically transformed to faceted GBs containing nearly ideal 30° < 0001> -GBs, which is driven by reducing GBs energy. Corresponding to the low energy and high mobility of both 30° < 0001> -GBs and 30° (0001)-GBs, these boundaries may dominate the grain growth phenomenon, leading to the growth of basally oriented grains, which results in the strengthening of the texture intensity.

Methods
Atomistic models including topological models and molecular static/dynamic simulations are employed to characterize structures and properties of < 0001> tilt and (0001) twist boundaries in Mg. Here we studied symmetric tilt and twist grain boundaries that are the simplest of < 0001> tilt and twist GBs and only required two parameters (the tilt/twist axis and tilt/twist angle 2θ) to describe their crystallographic relations 29,30 . For simplicity of the following description and discussion, < 0001> symmetrical tilt grain boundaries are referred to as < 0001> -GBs and < 0001> symmetrical twist grain boundaries are referred to as (0001)-GBs. We employ molecular dynamics (MD) simulations to characterize atomic structures of < 0001> -GBs and (0001)-GBs over the entire range of the rotation/twist angle θ. In the fixed coordinate system (Fig. S1a), the x-axis lies along [2110], the y-axis lies along [0110] , and the z-axis lies along [0001]. The two crystals originally adopt the same local coordinate system as the fixed coordinate system. For < 0001> -GBs, the z-axis is the tilt axis and the z-x plane is the GB plane. The two crystals rotate an angle θ clockwise and counterclockwise about the z-axis, respectively ( Figure S1b). The corresponding MD simulation cell containing a single GB on the z-x plane is illustrated in Figure S1c. Symmetrical twist grain boundaries are assembled using the same operation with the boundary plane on the x-y plane. The two crystals twist around the z-axis. In MD simulations, the simulation model contains two parts -a moveable region inside the simulation cell and a semi-rigid region that surrounds the movable region. The semi-rigid region acts as a flexible boundary to mimic the bulk response during MD relaxation 38 . For < 0001> -GBs, periodic boundary conditions are applied in both the x and z directions. The x dimension varies with respect to the tilt angle θ such that periodic boundary conditions are satisfied. The height of the two crystals in the y is 8.0 nm and the thickness of semi-rigid regions in the y is 1.2 nm, which is two times the cutoff of the potential used. The dimension in the z direction is ~2.6 nm. For (0001)-GBs, we adopted a cylindrical bi-crystal with the longitude axis along the z-axis and the radius of 10 nm. The height of the two crystals in the z-axis is 10.0 nm and the thickness of semi-rigid regions is 1.2 nm in the radial direction and in the z-axis (Fig. S1d).
The bi-crystal models are relaxed at 0 K by quenching molecular dynamics using an embedded atom method (EAM) potential for Mg 39 . This EAM potential reproduces well many experimentally measured properties and predictions of defect formation energies using first principle density function theory calculations. The predictor-corrector algorithm developed by Gear was used in our simulations with a temp step of 0.002 ps 40 .