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Grain-boundary activated pyramidal dislocations in nano-textured Mg by molecular dynamics simulation
Materials Science and Engineering A 528 (2011) 5411–5420
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Grain-boundary activated pyramidal dislocations in nano-textured Mg by molecular dynamics simulation D.-H. Kim a , F. Ebrahimi a,1 , M.V. Manuel a , J.S. Tulenko b , S.R. Phillpot a,∗ a b
Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611, United States Department of Nuclear and Radiological Engineering, University of Florida, Gainesville, FL 32611, United States
a r t i c l e
i n f o
Article history: Received 6 December 2010 Received in revised form 22 February 2011 Accepted 23 February 2011 Available online 3 March 2011 Keywords: Magnesium Dislocation Pyramidal slip Molecular dynamics
a b s t r a c t The generation and structures of first- and second-order pyramidal c + a dislocations, 1/3{1 0 1¯ 1} 1¯ 1¯ 2 3 and 1/3{1 1 2¯ 2} 1¯ 1¯ 2 3, are determined in pure magnesium using molecular dynamics simulation. In particular, simulations of [1 1 2¯ 0]- and [1 0 1¯ 0]-textured polycrystalline Mg display pyramidal c + a slip nucleated at grain boundaries. Both the first- and second-order dislocations appear as a partial or extended edge type. In the [1 1 2¯ 0]-textured Mg, the first-order pyramidal c + a slip occurs with 1/62¯ 0 2 3 partials or 1/9[0 1¯ 1 3] + 1/18[6¯ 2 4 3] + 1/6[0 2¯ 2 3] extended dislocations. Secondary pyramidal dislocations are created with edge type from grain boundaries in the [1 0 1¯ 0]-texture. The pyramidal c + a slip on the {1 1 2¯ 2} plane can extend to the basal plane, on which it is terminated by a screw dislocation on the {1 0 1¯ 1} plane. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Hexagonal close-packed (HCP) metals manifest either prismatic a or basal a slip as their principal slip mode [1]. While the primary a slip mode is prismatic in Ti, Zr, and Hf, basal slip dominates in Co, Zn, Be and Mg [2,3]. Usually, if the primary slip is basal, then the secondary slip mode is prismatic, and vice versa. These two slip modes can, however, only produce strain along the a directions. Hence, these two a slip systems alone are not enough for homogeneous plastic deformation [2]. Activation of pyramidal c + a slip enhances the homogeneity in plastic deformation of HCP metals by providing a strain component along the c direction. The pyramidal c + a slip in HCP crystals has two modes: a firstorder {1 0 1¯ 1} 1¯ 1¯ 2 3 mode and a second-order {1 1 2¯ 2} 1¯ 1¯ 2 3 mode. Such pyramidal c + a slip plays a key role in the ductility of polycrystalline HCP metals [4].The c + a dislocations have been studied in experiment [4–6] and by theory and simulation [3,7–15]. In the experiments, deformed polycrystalline samples were examined using an optical microscope (OM), or transmission electron microscope (TEM), in order to identify the c + a dislocations. It was found in Mg that {1 1 2¯ 2} 1¯ 1¯ 2 3 pyramidal dislocations are strongly bound on the basal planes [4]. Tonda and Ando [6] reported that c + a edge dislocations are immobilized by their dissociation
∗ Corresponding author. Fax: +1 352 392 7219. E-mail address: [email protected]fl.edu (S.R. Phillpot). 1 Deceased. 0921-5093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2011.02.082
into a sessile c and a glissile a dislocation in Cd, Zn and Mg. Simulation studies have focused on the structures and stability of the c + a dislocation. A {1 1 2¯ 2} 1¯ 1¯ 2 3 second-order pyramidal edge dislocation was shown by Minonishi et al. [11,12] and Liang and Bacon [9] to have two types of cores; undissociated and dissociated. Yoo et al. [3] found the {1 1 2¯ 2} 1¯ 1¯ 2 3 edge dislocation core with a Shockley partial connected by a basal-plane stacking fault to a sessile partial. They also suggested from a calculation using anisotropic elasticity that c + a screw dislocations can cross slip from a prismatic plane into a pyramidal plane [3]. Analysis [7] of simulations using a Lennard–Jones potential revealed a temperature dependence of the core structure in c + a edge and screw dislocations, and double cross-slip of c + a screw dislocation on {1 0 1¯ 1} and {1 1 2¯ 2}. In contrast to studies focusing on second-order pyramidal slip of {1 1 2¯ 2} 1¯ 1¯ 2 3, first-order pyramidal slip has been addressed in Ti [5] and Mg [8]. In particular, Numakura et al. [14,15] suggested three different core structures for {1 0 1¯ 1} 1¯ 1¯ 2 3 dislocations, one planar and two non-planar. Li and Ma [8] reported first-order pyramidal slip, {1 0 1¯ 1} 1¯ 1¯ 2 3 in Mg. In simulation studies, pyramidal dislocations have been created in single crystals either by manually moving some atoms [9–15] or by applying external stress to an artificial cavity [8]. It is not possible to investigate the initiation of c + a dislocations when they are constructed by hand. In creating a c + a dislocation from a cavity, the only source of the pyramidal dislocation is the cavity itself. It is thus hard to see various activation processes of c + a slip that might be active in polycrystalline materials, in which grain boundaries (GBs) can be a source of dislocations [16].
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Fig. 1. (a) The first- and second-order pyramidal slip planes in an HCP unit cell. (b) Columnar microstructure with [1 1 2¯ 0] or [1 0 1¯ 0] texture.
This study focuses on the heterogeneous nucleation of c + a dislocations from GBs in polycrystalline Mg. In particular, the structure and properties of both first- and second-order pyramidal c + a dislocations in Mg are compared.
2. Simulation approach All of the nanocrystalline structures simulated here consist of four columnar grains of 40 nm diameter. Each grain in the structure has the same crystallographic orientation along the zaxis; thus the grains are separated by tilt grain boundaries. Two different orientations are considered, as shown in Fig. 1: one with [1 1 2¯ 0] texture, the other with [1 0 1¯ 0] texture. The [1 1 2¯ 0] and [1 0 1¯ 0] textures have their z-axes parallel to the first- and second-order pyramidal c + a slip planes, {1 0 1¯ 1} and {1 1 2¯ 2}, respectively. In these textures, a hexagonal unit cell has twofold symmetry along the z-direction of the simulation cell, with the crystallographic c-axis parallel to the x–y plane. The interatomic interactions are described by an embedded atom method (EAM) potential [17]. The deformation simulations are conducted at 293 K with 3d-periodic boundary conditions applied to the simulation cell. The time step is t ≈ 0.4 fs. A uniaxial tensile load is applied along the x-direction in creep mode: the magnitude of the applied stress is constant within each simulation, with values ranging from 0.9 to 1.45 GPa. The Parrinello–Rahman constant stress algorithm [18] and the Nose–Hoover thermostat [19] are also implemented. The activation of specific dislocations at specific values of applied stress is determined from these creep tests. Common neighbor analysis (CNA) [20,21] is used to distinguish atoms in FCC-like stacking faults from HCP coordinated atoms. The ‘AtomEye’ software is used for visualization of MD results [22]. Complete details of the simulation set up are described in an earlier paper [23].
3. Simulation results The [1 1 2¯ 0]- and [1 0 1¯ 0]-textured structures manifest different pyramidal dislocations under tensile stress. The fundamentals of the shape and nucleation condition of each pyramidal dislocation are compared in this section. In addition, dislocation structures are analyzed in detail.
3.1. Occurrence of pyramidal dislocations As reported in an earlier paper [23], pyramidal c + a slip processes are common in the [1 1 2¯ 0] Mg columnar texture. To activate pyramidal slip in this work, the [1 1 2¯ 0]-texture is strained under uniaxial stress of 1.18 GPa. After activation of pyramidal dislocations, the structure is quenched at 0.01 K. The structure is characterized with CNA to distinguish FCC from HCP environments. Gray, brown and black denote atoms in HCP environments, FCC environments (in stacking faults) and non-twelvefold coordinated, i.e., non-HCP or FCC (in dislocations and grain boundaries). In the [1 1 2¯ 0]-textured structure, only partial dislocations are generated from GBs under low applied stresses. As discussed previously [23], the structure labeled ˛ in Fig. 2(a) is a 1/62 0 2¯ 3 pyramidal c + a dislocation. At higher applied stresses, a transition of the partial c + a dislocation into two different structures takes place. Fig. 2(b) and (c) shows the structures for a stress of 1.45 GPa. One of the regions of the c + a partials is of the extended type, as seen in Fig. 2(b). The extended c + a pyramidal dislocations are denoted ˛ . The other high stress region is denoted ˛ in Fig. 2(c). It seems not to be an extended dislocation, but rather the result of the connection of long and short partial pyramidal dislocations. In these simulations, the ˛ structure is generally produced at higher stress than the ˛ . Pyramidal c + a slip is also found in a simulation of the [1 0 1¯ 0]-textured microstructure at an applied stress of 1.24 GPa. In Fig. 2(d), ˇ and ˇ indicate partial and extended pyramidal c + a dislocations, respectively. However, c + a slip nucleation appears to be less frequent in the [1 0 1¯ 0]-texture than in the [1 1 2¯ 0]-texture. Moreover, there is a difference between c + a partials in the [1 1 2¯ 0]- and [1 0 1¯ 0]-textured polycrystals. The c + a partials can glide through a grain in the [1 1 2¯ 0] texture, via a stacking fault mechanism. However, after being activated in the [1 0 1¯ 0]-texture, the partial denoted as ˇ in Fig. 3(d) is unstable, and hence immediately transforms into an extended c + a dislocation.
3.2. Structures of c + a dislocations We first examine the partial and extended dislocations appearing in the [1 1 2¯ 0]-textured structure. To understand the mechanism of partial slip, the structure of the leading core connected to a staking fault is illustrated in Fig. 3; the c + a dislocation is seen as black atoms among the atoms in HCP environments (gray). Blue dots denote the positions of the atoms before slip.
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Fig. 2. Snapshots of pyramidal c + a slip activated in [1 1 2¯ 0]- and [1 0 1¯ 0]-textured structures. Gray, black and brown denote normal (HCP), disordered (non-HCP or FCC), and stacking fault (FCC) atoms, respectively. Arrows denote the directions in which the c + a dislocations move. (a) The partial c + a pyramidal (˛) and basal dislocations, shown as brown lines, are nucleated at 1.18 GPa. (b) Extended pyramidal dislocation, ˛ , is activated at 1.45 GPa. Its shape changes during glide. (c) Two connected partial dislocations, ˛ , appear at 1.45 GPa. (d) Pyramidal c + a dislocations with partial (ˇ) and extended types (ˇ ) are found during deformation in [1 0 1¯ 0]-textured Mg. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The HCP stacking sequence in a basal direction is represented as alternating ‘A’ and ‘B’ layers. Green arrows denote directions of the shear stress that causes the slip. In the non-basal SF, the atomic displacement, indicated by the yellow arrows is parallel to [1 1¯ 0 2] and normal to [1 1 2¯ 0]. These two directions are on the {1 0 1¯ 1} plane between two c + a pyramidal slip planes previously illustrated at Fig. 1(a); hence, the depicted dislocation is a first-order pyramidal partial. The observed dislocation has a partial core and staking fault (SF), as shown in Fig. 3(b). The pyramidal c + a dislocation line exists in the [1 1 2¯ 0] direction, indicated as ‘⊥’ in Fig. 3(b). The potential energy (PE) of atoms in dislocation structure is also shown in Fig. 3(b). Although the atom positions show some disorder at the core of the partial dislocation, a stacking fault is formed by coordinated atomic motion. The two layers of the stacking fault colored in black move in opposite directions: one along 1/12[2 0 2¯ 3], ¯ These opposite atomic displacements the other along 1/12[2¯ 0 2 3]. ¯ partial slip, as shown previously combine to produce 1/6[2 0 2 3] [23]. An extended pyramidal dislocation in the [1 1 2¯ 0]-texture is more complicated than the partial dislocation. CNA results and PEs of the extended dislocation are shown in Fig. 4(a) and (b), respectively. The extended dislocation can be divided into leading and trailing partials connected with a SF, see Fig. 4(a). The regions with high PE in Fig. 4(b) correspond to the leading and trailing partial cores. Extra-half planes are found near the cores. There is a boundary of layers, denoted by the dotted (blue) line, at the upper side of the extended dislocation. The nature of this boundary will be dis-
cussed below in connection with sequential slip that leads to the formation of an extended dislocation. The structures of the leading and trailing partials are characterized in detail in Fig. 5. Among the three stacking fault lines (layers I, II, and III) of the trailing core, layers II and III move in opposite directions, as indicated by the yellow arrows in Fig. 5(a). To quantitatively determine their displacements, the atomic structure of layers I, II, and III are illustrated in Fig. 5(c). The slip of layers II ¯ They thus form a total of and III is 1/12[2 0 2¯ 3] and 1/12[2¯ 0 2 3]. 1/6[2 0 2¯ 3] partial slip, in the same manner as shown earlier in Fig. 3. The atomic structure of the dislocation core in the leading partial is also exhibited in Fig. 5(b). Each atom moves in the direction denoted by the yellow arrows. Some of atomic displacements are not parallel to the glide direction of the extended pyramidal dislocation in the leading core of Fig. 5(d). In the leading core, all the atomic displacements collectively make two components of partial slip: 1/18[4 2 6¯ 3] and 1/9[1 1¯ 0 3]. Fig. 6 shows the Burgers vectors of the extended pyramidal dislocation in the HCP unit cell. The three partials observed in the leading and trailing parts produce perfect pyramidal slip as follows: C + B0 + B0 B → CB 1/9[0 1¯ 1 3] + 1/18[6¯ 2 4 3] + 1/6[0 2¯ 2 3] → 1/3[1¯ 1¯ 2 3]
(1)
In the above, it has been demonstrated that the c + a dislocations in the [1 1 2¯ 0]-textured structure lie on the first-order
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Fig. 4. CNA and PE map of the extended pyramidal dislocation observed in the [1 1 2¯ 0]-textured structure. (a) ˛ denotes the extended c + a dislocation. Blue dots indicate a boundary of different layers: A and B denote different layer in the direction of [1 1 2¯ 0]. (b) The white ‘⊥’ represent the cores of the edge dislocations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Common neighbor analysis (CNA) and potential energy (PE) map of the partial pyramidal dislocation in the [1 1 2¯ 0]-textured structure. The white ‘⊥’ denote the cores of the edge dislocations in (b).
pyramidal plane, appearing either as partials or as fully extended dislocations. Now we focus on the pyramidal dislocations found in the [1 0 1¯ 0]-texture. In order to characterize their structure, the atomic-level details of the c + a dislocation found in the [1 0 1¯ 0] texture of Fig. 2 are shown in Fig. 7. The CNA result in Fig. 7(a) shows that the dislocation has an extra half-plane parallel to the basal direction. The dislocation seems to be divided into two different parts, as seen in the potential energy map of Fig. 7(b). The head partial, which contains a series of high-energy atoms, lies parallel to the z-direction, [1 0 1¯ 0]. The left image of Fig. 7(b) shows that the head core spreads into the basal plane. This basal plane connects to the tail part, seen as an angular plane. This shape is quite different from the extended pyramidal dislocation formed in the [1 1 2¯ 0]-textured structure. Because the structures of the head and tail partials are keys to understanding a formation mechanism of
the extended c + a dislocation, they are displayed in further detail in Fig. 8. Fig. 8 shows CNA images projected on both the (1 0 1¯ 0) and (0 0 0 1) planes in the two left columns, while the potential energy (PE) on (0 0 0 1) is mapped in the right column. The firstand second-order pyramidal c + a slip planes are seen as semitransparent red and blue. Blue dots and yellow arrows denote the sites of atoms before slip and the atomic displacement, respectively. The stacking sequence of HCP in the basal direction is also seen as ‘A’ and ‘B’. Slip on the secondary c + a plane can be seen in the CNA results of Fig. 8(a). Referring to the (0 0 0 1) oriented CNA figure, the yellow arrows indicate that the atoms do not displace in the same direction as in secondary pyramidal slip. The difference in the slip behavior at each atomic layer parallel to the slip plane is due to the motion of zonal dislocations known to typically occur in pyramidal slip [1]. The zonal dislocations appear as zigzag ‘shuffling’ displacements. They are also known to make displacement of ¯ see Fig. 6 [1]. Most of the atomic displacements ∼1/4B C ([1 1 2¯ 3]), shown in the vicinity of secondary pyramidal slip plane, {1 1 2¯ 2}, of Fig. 8 are also ∼1/4B C. In addition, the PEs of atoms on the slip plane are higher than those of bulk HCP atoms. The next basal layer of the cross section of Fig. 8(a) along is shown in Fig. 8(b). The secondary pyramidal slip has progressed more in Fig. 8(b), as evidenced by atomic displacements and PEs on the slip plane of Fig. 8(b). As shown in the (0 0 0 1)-oriented CNA result of Fig. 8(a), the atomic displacement takes place on one layer parallel to the secondary c + a slip plane of (1 1 2¯ 2); yellow arrows form a monolayer in the [1 1 2¯ 0] microstructure. Two layers move, however, by individual displacement of atoms adjacent to the slip plane, as shown by the two yellow arrows in the (0 0 0 1)-oriented CNA result of Fig. 8(b).
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Fig. 5. Atomic structures of a (a) leading and (b) trailing of the extended c + a dislocation including CNA. (c) and (d) corresponding atomic displacement on the first-order pyramidal slip plane, (1 0 1¯ 1).
As the secondary pyramidal slip progress, a region of atoms with high PEs appears between the secondary pyramidal slip plane and an angular boundary in the (0 0 0 1)-oriented PE map of Fig. 8(b).
Fig. 6. Burgers vectors of the first-order pyramidal extended dislocation shown in the HCP unit cell.
This implies that additional displacements occur on a basal plane by the motions of zonal dislocations on a second pyramidal plane. The stress on the basal plane thus activates slip on the first-order pyramidal plane of (1 0 1¯ 1), as exhibited in Fig. 8(c). The extended pyramidal dislocation is thus revealed to contain both first- and second-order pyramidal slip. As already observed in Fig. 7(a), the secondary pyramidal slip is edge type, as evidenced by an extra half plane of atoms. However, because the first-order c + a partial slip does not show an extra-half plane, it appears to be a screw dislocation. Fig. 9 exhibits the tail partials of the extended c + a dislocation structure in the [1 0 1¯ 0] texture. The semi-transparent red and blue sheets denote the first- and second-order pyramidal slip planes, respectively. The atomic displacements associated with slip are represented by green arrows. The edge dislocation line of the secondary c + a slip is denoted te in Fig. 9(a) and (b). The first-order pyramidal slip is initiated at Fig. 9(b). As shown in the (0 0 0 1)oriented CNA result of Fig. 8(c), the atomic motions occur along 1 1 2¯ 0. However, as Fig. 9(c) shows, the atomic motions change to the pyramidal c + a direction. The presence of a screw dislocation can be seen in Fig. 9(g) and (h). Although the same plane is seen in both, in Fig. 9(g) line ˛ is disconnected, and line ˇ is continued, while in Fig. 9(h) the line ˛ is continued, and line ˇ is disconnected. This feature is typical of screw dislocations. Hence, the pyramidal dislocation at the trail partial is screw type, having a dislocation line, ts .
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Fig. 7. (a) CNA and (b) PE map of the pyramidal c + a dislocation activated in the [1 0 1¯ 0]-textured structure.
Fig. 8. CNA visualization and PE map of the extended c + a pyramidal dislocation projected onto the (1 0 1¯ 0) and (0 0 0 1) planes. Atomic structures where one and two layers of top basal surface are removed at (a) are shown at (b) and (c), respectively. ‘A’ and ‘B’ represent the stacking sequence of HCP in the basal direction. The shear direction is marked by the green arrow. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 9. Layer structures of the extended pyramidal c + a dislocation in the [1 0 1¯ 0]-texture by CNA. The semi-transparent red and blue sheets denote the first- and secondorder pyramidal slip planes, respectively. ‘A’ and ‘B’ represent the stacking sequence of HCP in the basal direction. The direction of local slip is seen as a green arrow. te and ts indicate a dislocation line of screw and edge, respectively. ˛˛ and ˇˇ denote discontinuous atomic layers in the basal direction. However, each atom has less level difference ˚ in continuous lines of ˛ or ˇ. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this than atomic radius (1.6 A) article.)
4. Discussion Comparing the [1 1 2¯ 0]- and [1 0 1¯ 0]-textured structures, there are some significant points to be addressed with regards to their c + a slip behaviors. In this section, the previously presented results are analyzed in terms of c + a dislocation structures. The differences in the formation and properties of first- and secondorder pyramidal dislocations are discussed.
1/6[2¯ 0 2 3] + 1/6[0 2¯ 2 3] → 1/3[1¯ 1¯ 2 3]
In the above analyses, both first-order [5,8,9,14,15] and secondorder [4,6,7,10–13] pyramidal dislocation structures have been considered. For the first-order pyramidal plane, Liang and Bacon [9] studied the pyramidal slip on the {1 0 1¯ 1} plane using a Lennard–Jones potential. They identified a dissociation reaction:
(5)
Morris et al. [13] suggested a Type III variant, in which the dissociation is different: 1/3[1 0 1¯ 0] + 1/3[0 1 1¯ 3] → 1/3[1¯ 1¯ 2 3]
4.1. Dislocation structures
1/4[1¯ 0 1 2] + 1/12[1 4¯ 3 2] + 1/6[1 0 1 2] → 1/3[1¯ 1¯ 2 3]
The secondary c + a edge dislocation is known to display three variants [11,13]. Type I is a perfect dislocation, while Type II consists of two 1/2c + a partials. The Type I transforms to the Type II on heating from 0 K to 293 K [7]. The dissociation reaction of Type II is:
(6)
Partial edge cores of Type II are found in the [1 0 1¯ 0]-textured structure in our simulations. In contrast to these reactions, pyramidal dislocations in the [1 0 1¯ 0]-textured structure have both edge and screw components. Considering a screw dislocation, there are two issues to be compared to previous studies. First, the screw dislocation is likely to cross-slip: it has been reported that a c + a
(2)
Li and Ma suggested a dissociation process of the first-order c + a dislocation in Mg: 1/6[2 1¯ 1 0] + 1/6[0 1 1¯ 3] + 1/6[0 2 2 3] → 1/3[1¯ 1¯ 2 3]
(3)
Although their Mg EAM potential [17] is the same as that used in this work, the reaction in Eq. (3) is quite different from that seen in our simulations: 1/9[1¯ 0 1 3] + 1/18[2 6¯ 4 3] + 1/6[2¯ 0 2 3] → 1/3[1¯ 1¯ 2 3]
(4)
This mechanism was previously reported by Jones and Hutchinson [5]. Using a hard sphere model for a titanium alloy (Ti–6Al–4V), they characterized the first-order pyramidal slip as dissociated. Thus, at this point, there is a question as to why the same dissociation process takes place in both Mg and Ti. While Mg has an essentially spatially isotropic electronic structure of s and p orbitals [23], the 3d2 orbitals of Ti make it strongly anisotropic [23]. Such an anisotropy is responsible for the non-ideality in the c/a ratio and differences in the dominant slip mode: Mg and Ti have basal and prismatic slip as a dominant slip mode, respectively [24]. It is important to note that the work by Jones and Hutchinson [5] was performed using an intrinsically isotropic hard-sphere model. Hence, it is not surprising that the partial dissociations they observed are the same as those we see in Mg.
Fig. 10. Potential energy distributions (snapshots) and strain energy curves within cylinders of radius R having a center of an edge pyramidal c + a dislocation line.
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Fig. 11. Pyramidal c + a dislocations simulated by Sun potential. (a) CNA image of [1 1 2¯ 0] texture at 1.45 GPa; black, brown and cyan denote disordered (non-HCP or FCC), stacking fault (FCC) atoms, and twinned area. ˛1 is connected with FCC stacking faults, and ˛2 shows two c + a dislocations joined on two different slip planes. (b) Potential energy analysis of [1 0 1¯ 0] texture at 1.32 GPa in (a). Arrow indicates the directions in which the c + a dislocations move ˛ and ˇ denote pyramidal c + a and basal a dislocations. ˛ has the same structure as the extended c + a dislocation illustrated in Fig. 7. An intergranular crack appears between grains 2 and 4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
screw dislocation spreads on two {1 0 1¯ 0} first-order pyramidal planes at 0 K [7]. Yoo et al. [3] concluded that a c + a screw dislocation may split on multiple slip planes, e.g. {1 0 1¯ 0} and {1 1 2¯ 2}. In the [1 0 1¯ 0]-textured structure the pyramidal c + a dislocation takes place by cross-slip on the first- and second-order slip planes mediating a basal plane (see Figs. 8 and 9). The screw core also appears as a junction of two {1 0 1¯ 0} planes in this simulation. Second, the secondary c + a edge dislocation becomes immobilized: the secondary pyramidal edge dislocations of Type II is known to be sessile at >30 K because of the extended core along the (0 0 0 1) [7]. Type III is also sessile with a basal stacking fault, as seen in Eq. (6) [13]. This spreading of the second-order pyramidal dislocation on the basal plane was already observed in Figs. 7–9. This non-planar core spreading in HCP causes dislocations to be sessile [24]. The immobilization of secondary c + a edge dislocations may be one of the reasons why it is difficult to form long partial c + a dislocations in the [1 0 1¯ 0]-texture, as previously mentioned in Section 3.1. 4.2. Activation of first- and second-order c + a slip The simulations described above show that c + a pyramidal slip processes are more easily observed in the [1 1 2¯ 0]-texture than in the [1 0 1¯ 0]-texture. Hence, first-order pyramidal slip in the [1 1 2¯ 0]-texture takes place more easily than second-order
pyramidal slip in the [1 0 1¯ 0]-texture. This is in disagreement with the experimental findings that secondary pyramidal slip on 1/3{1 1 2¯ 2} 1¯ 1¯ 2 3 dominates in Mg. To address this discrepancy, we compare the atomic motion, potential energies, and stacking fault energies of pyramidal edge dislocations in the two textures. In the [1 1 2¯ 0]-texture the first-order partial pyramidal dislocation, 1/6[2 0 2¯ 3], arises from partial motion of two adjacent layers, ¯ as previously shown in Fig. 3. ∼1/12[2 0 2¯ 3] and 1/12[2¯ 0 2 3], By contrast, the core of secondary pyramidal slip with edge type is obtained by full motion of only one layer with zonal behavior in Fig. 8. For the purposes of an energy analysis of the firstand second-order pyramidal dislocations, the distributions of total strain energy per Burgers vector are represented in Fig. 10 as a function of distance from the centers of dislocation cores. The total strain energy (Etotal ), expressed by adding core and elastic strain energy (Ecore + Eelastic ) [25], is useful for analyzing the size and energy of a dislocation core. Pyramidal dislocations are partial or extended types in this work: the 1st and 2nd pyramidal dislocations are connected with a {1 0 1¯ 1} SF and with screw dislocation consisting of two {1 0 1¯ 1} planes, respectively. In order to focus on the edge cores of pyramidal slip, we examine the total strain energy of core regions (Ecore ) rather than that of the elastic region (Eelastic ). The magnitude of Burgers vector (b) for both partial pyramidal slip ˚ the total strain energy is plotted to 3.75b, as shown in is 3.2 A;
Table 1 Stacking fault energies in magnesium. Stacking fault energy (Esf , mJ/m2 ) Basal a, {0 0 0 1} ISF-1
First-order c + a slip plane, {1 0 1¯ 1}
Method Second-order c + a slip plane, {1 1 2¯ 2}
ISF-2
This work
24
48
122
253
MD (poly), Liu EAM
Nogaret [29]
27 – –
54 44 –
121, 225 118, 240 180
198 × 236
MD (single), Liu EAM MD (single), Sun EAM Ab initio
Morris [30] Chetty [31] Jelinek [32] Yasi [33] Wang [34]
– – 18 –
– 44 37 34 21
– – – –
173 – – –
– – – –
78 60 >90 <50
– – – –
– – – –
Sastry [35] Devlin [36] Fleischer [37] Couret [38]
Experiment
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of basal intrinsic stacking fault-1 (ISF-1) and -2 (ISF-2) are less than those which were reported by Liu [17]. This discrepancy may be caused by imperfect relaxation of our polycrystalline structure due to a residual stress after partial slip process; by contrast single crystals having no other defects would allow complete stress relaxation. However, our SFEs are still in the range of values experimentally reported. As shown in Table 1, the SFE on the second-order c + a slip plane, 253 mJ/m2 is twice that of the SFE on the first-order c + a slip plane, 122 mJ/m2 . The second-order pyramidal c + a dislocations did not show long SFs connecting their partial cores, whereas long SFs are easily produced in the first-order c + a dislocations. Presumably, this is a result of the differences in their SFEs. In summary, the atomic motions and partial core energies explain the more frequent occurrence of the first-order pyramidal dislocation. Although the second-order c + a edge dislocations nucleate GBs, their non-planar structure prevent them from gliding inside a grain. As local stress is concentrated at these sessile dislocations, however, the nucleated partials become mobile. Due to their high SFEs, they normally appear as extended dislocations rather than partial cores separated by long SFs. Moreover, the high SFE of second pyramidal c + a slip indicates that the second-order c + a slip are likely to be a screw type or dissociated (c + a → sessile c + glissile a) processes rather than pure edge type, as reported in previous studies [4]. It is instructive to compare these simulation results with experimental findings. Tonda and Ando [6] reported a difference in yield shear stress between first- and second-order pyramidal slip of only ∼20%. This difference is small comparable to the critical resolves shear stress (CRSS) ratio for non-basal to basal slip. This indicates that first-order pyramidal slip of 1/3{1 0 1¯ 1} 1¯ 1¯ 2 3 should also be manifested for appropriate textures and applied stresses; activation of deformation modes in HCP metals, such as Mg, strongly depends on texture and applied stress [39]. Indeed, Li and Ma [8] also suggested that first-order pyramidal slip takes place in MD simulation of single crystalline Mg, and identified the associated stacking faults in TEM [40].
4.3. Comparison of different EAM potentials
Fig. 12. Creep curves and snapshots of strained columnar Mg with 18 nm grain size. Cyan, brown and black in (b) and (c) mean a twinned region, stacking fault (FCC), disordered (non-HCP or FCC), respectively. (a) Time–strain curves at 1.035 and 1.18 GPa. (b) Snapshot at ˛ of 1.18 Gpa-curve. (c) Snapshot at ˇ of 1.035 Gpacurve. A red circle indicates a region where an interaction between twinning and c + a dislocations take place. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 10. Taking the dislocation core size to be 3b [25–28], it is possible to estimate the core energy from Fig. 10. The second-order pyramidal dislocation core in the [1 0 1¯ 0]-texture is found to have much higher strain energy than the first-order pyramidal slip in the [1 1 2¯ 0]-texture. In addition, analysis of stacking faults energies (SFEs) in Mg from various simulation and experimental methods are compared in Table 1. The SFEs in the present work were determined from SF regions >5 nm apart from a core, created in polycrystalline by partial slip. Since there has been little analysis of SFEs on the c + a plane, we tried to obtain consistency of our results through comparing well-known SFEs on the basal plane. The calculated energies
The properties of a material as determined in a simulation can depend strongly on the description of the interatomic interactions. There are two literature EAM potentials for Mg: one developed by Liu et al. [17], the other developed by Sun et al. [41]. To this point, we have adopted the Liu potential to simulate plastic deformation of nanocrystalline Mg [23]. The EAM potential fitted by Sun et al. has been found to manifest more active basal slip behavior than the Liu potential [42], a result of the lower Peierls stress [29]. We have compared the deformation behavior as given by the Sun potential with the results above for the Liu potential. We find that there are far fewer nucleation events for twinning and c + a slip for the Sun potential [42]. In addition, intergranular cracks are frequently seen even at small plastic strains (<3%) during plastic deformation process of polycrystalline sample, as shown in Fig. 11(b), presumably a consequence of the reduced twinning and c + a slip activity. Such cracks are common in simulation of 3D polycrystalline Mg using Sun potential [42]. As described by the Liu potential, the first-order pyramidal c + a partial slip in the [1 1 2¯ 0] texture shows partial dislocations joined by long SFs; moreover the extended c + a dislocations glide easily. In contrast, Sun potential produces comparatively short non-basal SFs often connected with a SF of FCC on the basal plane, as exhibited in Fig. 11(a), and less mobile extended c + a dislocations. Comparing the structure of c + a dislocations using Liu and Sun potentials in [1 1 2¯ 0] and [1 0 1¯ 0] textures of Mg, there was no significant difference (Fig. 11), consistent with the results of [29].
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4.4. Role of pyramidal c + a slip in a plastic deformation The pyramidal c + a dislocations are found to play a number of roles. First, homogeneous plastic deformation requires pyramidal c + a slip [43,44]. A basal or prismatic mode appears as dominant slip in HCP metals. However, if some grains require high CRSS to activate the basal or prismatic mode, the secondary slip mode, pyramidal c + a, should be activated for plastic deformation. Activation of this secondary mode has a direct effect on mechanical response of the material at meso- and macro-levels. Single crystalline textures of Mg show noticeable differences in stress–strain curve according to the directions in which stress is applied [45]. The activation of secondary c + a slip is one of the important reasons for these differing deformation responses. Second, the interaction between c + a slip and twinning results in hardening. The twinning and c + a slip compete in grains in which basal slip is rare. Fig. 12 shows hardening from the interaction between the c + a dislocation and the twinning boundary. Initially the simulated [1 1 2¯ 0] textures are defect-free, except for the GBs. The interaction can be thus examined between dislocations and twinning created under the external stress. To maintain the specific stress value needed for the c + a slip–twinning interaction, a creep test is more suitable than a tensile test. The initial textures are strained at 1.035 and 1.18 GPa, showing a c + a dislocation–twinning interaction and only twinning in Fig. 12(b) and (c), respectively. While the 1.18 GPa-curve shows the strain rate (εA ) of 1.5 × 109 s−1 , a significant decrease (εB = 1.4 × 108 s−1 ) of stain rate appears in 1.035 GPa-curve of Fig. 12(a). As shown in Fig. 12 (b) and (c). We attribute this hardening to growth of the {1 0 1¯ 2} 1 0 1¯ 1 tensile twinning blocked by the c + a dislocations of order {1 0 1¯ 1} 1¯ 1¯ 2 3. In addition, it was experimentally reported that under compressive stress the c + a slip has a harmful effect on room temperature ductility of Mg: it produces severe strain hardening by dissociation to mobile a and immobile c dislocations [4,46]. 5. Conclusions Pyramidal c + a dislocations were created at GBs in [1 1 2¯ 0]and [1 0 1¯ 0]-textured Mg using MD simulation. It was found that both the first- and second-order pyramidal slip can be activated in Mg. The first-order c + a dislocations in the [1 1 2¯ 0]-texture manifested various forms, depending on the external stress. At low stress they are 1/62¯ 0 2 3 partials. Two phases, extended dislocation and connected partials, appear at high stress. The extended type of the first-order pyramidal slip takes place through a dissociation reaction of 1/9[0 1¯ 1 3] + 1/18[6¯ 2 4 3] + 1/6[0 2¯ 2 3] → 1/3[1¯ 1¯ 2 3]. While all of the dislocations in the [1 1 2¯ 0] texture are of edge type, the extended secondary pyramidal dislocation in the [1 0 1¯ 0] texture exhibits both types of edge and screw. The edge partial appears as a leading core of the secondary c + a slip, formed by typical zonal dislocation in HCP metals. The partial core, due to spreading on the basal plane and high SFE, remain sessile. During deformation, stress concentration which occurs at the sessile dislocation finally causes an activation of screw dislocation on the basal plane connected with the edge core. Hence, the complex dislocation is possible to glide in the [1 0 1¯ 0] columnar textures without leaving SF having high energy. Comparing the two different EAM potentials, the atomistic structures of pyramidal c + a dislocations are found to be very similar, though the Liu potential shows more active c + a slip than Sun potential. The textured columnar nano-structures considered here are in many ways ideal for fundamental studies; however, they are opti-
mized for edge dislocation. As a next step, large-scale randomly oriented, non-textured Mg polycrystals will be simulated; these will allow for a greater diversity of plastic process and allow more direct comparisons with experiment. Acknowledgments This work was supported by DOE NERI contract DE-FC0707ID14833. The work of FE was supported by the NSF under grant number DMR-0605406. The work of MVM was sponsored by the NSF Award Number DMR-0845868. References [1] J.P. Hirth, J. Lothe, Theory of Dislocations, Wiley, New York, 1982. [2] S. Balasubramanian, L. Anand, Acta Mater. 50 (2002) 133–148. [3] M.H. Yoo, J.R. Morris, K.M. Ho, S.R. Agnew, Metall. Mater. Trans. A 33 (2002) 813–822. [4] T. Obara, H. Yoshinga, S. Morozumi, Acta Metall. 21 (1973) 845–853. [5] I.P. Jones, W.B. Hutchinson, Acta Metall. 29 (1981) 951–968. [6] H. Tonda, S. Ando, Metall. Mater. Trans. A 33 (2002) 831–836. [7] S. Ando, T. Gotoh, H. Tonda, Metall. Mater. Trans. A 33 (2002) 823–829. [8] B. Li, E. Ma, Philos. Mag. 89 (2009) 1223–1235. [9] M.H. Liang, D.J. Bacon, Philos. Mag. A 53 (1986) 181–204. [10] Y. Minonishi, S. Ishioka, M. Koiwa, S. Morozumi, Philos. Mag. A 46 (1982) 761–770. [11] Y. Minonishi, S. Ishioka, M. Koiwa, S. Morozumi, M. Yamaguchi, Philos. Mag. A 43 (1981) 1017–1026. [12] Y. Minonishi, S. Ishioka, M. Koiwa, S. Morozumi, M. Yamaguchi, Philos. Mag. A 45 (1982) 835–850. [13] J.R. Morris, K.M. Ho, K.Y. Chen, G. Rengarajan, M.H. Yoo, Model. Simul. Mater. Sci. Eng. 8 (2000) 25–35. [14] H. Numakura, Y. Minonishi, M. Koiwa, Philos. Mag. A 62 (1990) 525–543. [15] H. Numakura, Y. Minonishi, M. Koiwa, Philos. Mag. A 62 (1990) 545–556. [16] A. Jain, S.R. Agnew, Mater. Sci. Eng. A 462 (2007) 29–36. [17] X.Y. Liu, J.B. Adams, F. Ercolessi, J.A. Moriarty, Model. Simul. Mater. Sci. Eng. 4 (1996) 293–303. [18] M. Parrinello, A. Rahman, J. Appl. Phys. 52 (1981) 7182–7190. [19] D.J. Evans, B.L. Holian, J. Chem. Phys. 83 (1985) 4069–4074. [20] A.S. Clarke, H. Jonsson, Phys. Rev. E 47 (1993) 3975–3984. [21] H. Jonsson, H.C. Andersen, Phys. Rev. Lett. 60 (1988) 2295–2298. [22] J. Li, Model. Simul. Mater. Sci. Eng. 11 (2003) 173–177. [23] D.H. Kim, M.V. Manuel, F. Ebrahimi, J.S. Tulenko, S.R. Phillpot, Acta Mater. 58 (2010) 6217–6229. [24] D.J. Bacon, V. Vitek, Metall. Mater. Trans. A 33 (2002) 721–733. [25] D. Hull, D.J. Bacon, Introduction to Dislocations, Oxford, Boston, Butterworth–Heinemann, 2001. [26] Y.N. Osetsky, D.J. Bacon, Model. Simul. Mater. Sci. Eng. 11 (2003) 427– 446. [27] R.E. Voskoboinikov, Y.N. Osetsky, D.J. Bacon, Mater. Sci. Eng. A 400 (2005) 45–48. [28] G.F. Wang, A. Strachan, T. Cagin, W.A. Goddard, Model. Simul. Mater. Sci. Eng. 12 (2004) S371–S389. [29] T. Nogaret, W.A. Curtin, J.A. Yasi, L.G. Hector, D.R. Trinkle, Acta Mater. 58 (2010) 4332–4343. [30] J.R. Morris, J. Scharff, K.M. Ho, D.E. Turner, Y.Y. Ye, et al., Philos. Mag. A 76 (1997) 76. [31] N. Chetty, M. Weinert, Phys. Rev. B 56 (1997) 10844. [32] B. Jelinek, J. Houze, S. Kim, M.F. Horstemeyer, M.I. Baskes, et al., Phys. Rev. B (2007) 75. [33] J.A. Yasi, T. Nogaret, D.R. Trinkle, Y. Qi, L.G. Hector, et al., Model. Simul. Mater. Sci. Eng. (2009) 17. [34] Wang R, Wang SF, Wu XZ, Wei QY. Phys Scripta 81. [35] D.H. Sastry, Y.V.R. Prasad, K.I. Vasu, Scripta Metall 3 (1969) 927. [36] J.F. Devlin, J. Phys. F 4 (1974) 1865. [37] R.L. Fleischer, Scripta Metall 20 (1986) 223. [38] A. Couret, D. Caillard, Acta. Metall. 33 (1985) 1455. [39] E.W. Kelly, W.F. Hosford, Trans. Metall. Soc. AIME 242 (1968) 654. [40] B. Li, P.F. Yan, M.L. Sui, E. Ma, Acta Mater. 58 (2010) 173–179. [41] D.Y. Sun, M.I. Mendelev, C.A. Becker, K. Kudin, T. Haxhimali, M. Asta, J.J. Hoyt, A. Karma, D.J. Srolovitz, Phys. Rev. B 73 (2006) 024116. [42] D.H. Kim, M.V. Manuel, F. Ebrahimi, J.S. Tulenko, S.R. Phillpot, 2010, unpublished work. [43] M.H. Yoo, S.R. Agnew, J.R. Morris, K.M. Ho, Mater. Sci. Eng. A 319–321 (2001) 87–92. [44] J. Koike, T. Kobayashi, T. Mukai, H. Watanabe, M. Suzuki, K. Maruyama, K. Higash, Acta Mater. 51 (2003) 2055–2065. [45] E.W. Kelley, W.F. Hosford Jr., Trans. AIME 242 (1968) 654–660. [46] J. Koike, Metall. Mater. Trans. A 36 (2005) 1689–1696.
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Materials Science and Engineering A journal homepage: www.elsevier.com/locate/msea
Grain-boundary activated pyramidal dislocations in nano-textured Mg by molecular dynamics simulation D.-H. Kim a , F. Ebrahimi a,1 , M.V. Manuel a , J.S. Tulenko b , S.R. Phillpot a,∗ a b
Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611, United States Department of Nuclear and Radiological Engineering, University of Florida, Gainesville, FL 32611, United States
a r t i c l e
i n f o
Article history: Received 6 December 2010 Received in revised form 22 February 2011 Accepted 23 February 2011 Available online 3 March 2011 Keywords: Magnesium Dislocation Pyramidal slip Molecular dynamics
a b s t r a c t The generation and structures of first- and second-order pyramidal c + a dislocations, 1/3{1 0 1¯ 1} 1¯ 1¯ 2 3 and 1/3{1 1 2¯ 2} 1¯ 1¯ 2 3, are determined in pure magnesium using molecular dynamics simulation. In particular, simulations of [1 1 2¯ 0]- and [1 0 1¯ 0]-textured polycrystalline Mg display pyramidal c + a slip nucleated at grain boundaries. Both the first- and second-order dislocations appear as a partial or extended edge type. In the [1 1 2¯ 0]-textured Mg, the first-order pyramidal c + a slip occurs with 1/62¯ 0 2 3 partials or 1/9[0 1¯ 1 3] + 1/18[6¯ 2 4 3] + 1/6[0 2¯ 2 3] extended dislocations. Secondary pyramidal dislocations are created with edge type from grain boundaries in the [1 0 1¯ 0]-texture. The pyramidal c + a slip on the {1 1 2¯ 2} plane can extend to the basal plane, on which it is terminated by a screw dislocation on the {1 0 1¯ 1} plane. © 2011 Elsevier B.V. All rights reserved.
1. Introduction Hexagonal close-packed (HCP) metals manifest either prismatic a or basal a slip as their principal slip mode [1]. While the primary a slip mode is prismatic in Ti, Zr, and Hf, basal slip dominates in Co, Zn, Be and Mg [2,3]. Usually, if the primary slip is basal, then the secondary slip mode is prismatic, and vice versa. These two slip modes can, however, only produce strain along the a directions. Hence, these two a slip systems alone are not enough for homogeneous plastic deformation [2]. Activation of pyramidal c + a slip enhances the homogeneity in plastic deformation of HCP metals by providing a strain component along the c direction. The pyramidal c + a slip in HCP crystals has two modes: a firstorder {1 0 1¯ 1} 1¯ 1¯ 2 3 mode and a second-order {1 1 2¯ 2} 1¯ 1¯ 2 3 mode. Such pyramidal c + a slip plays a key role in the ductility of polycrystalline HCP metals [4].The c + a dislocations have been studied in experiment [4–6] and by theory and simulation [3,7–15]. In the experiments, deformed polycrystalline samples were examined using an optical microscope (OM), or transmission electron microscope (TEM), in order to identify the c + a dislocations. It was found in Mg that {1 1 2¯ 2} 1¯ 1¯ 2 3 pyramidal dislocations are strongly bound on the basal planes [4]. Tonda and Ando [6] reported that c + a edge dislocations are immobilized by their dissociation
∗ Corresponding author. Fax: +1 352 392 7219. E-mail address: [email protected]fl.edu (S.R. Phillpot). 1 Deceased. 0921-5093/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2011.02.082
into a sessile c and a glissile a dislocation in Cd, Zn and Mg. Simulation studies have focused on the structures and stability of the c + a dislocation. A {1 1 2¯ 2} 1¯ 1¯ 2 3 second-order pyramidal edge dislocation was shown by Minonishi et al. [11,12] and Liang and Bacon [9] to have two types of cores; undissociated and dissociated. Yoo et al. [3] found the {1 1 2¯ 2} 1¯ 1¯ 2 3 edge dislocation core with a Shockley partial connected by a basal-plane stacking fault to a sessile partial. They also suggested from a calculation using anisotropic elasticity that c + a screw dislocations can cross slip from a prismatic plane into a pyramidal plane [3]. Analysis [7] of simulations using a Lennard–Jones potential revealed a temperature dependence of the core structure in c + a edge and screw dislocations, and double cross-slip of c + a screw dislocation on {1 0 1¯ 1} and {1 1 2¯ 2}. In contrast to studies focusing on second-order pyramidal slip of {1 1 2¯ 2} 1¯ 1¯ 2 3, first-order pyramidal slip has been addressed in Ti [5] and Mg [8]. In particular, Numakura et al. [14,15] suggested three different core structures for {1 0 1¯ 1} 1¯ 1¯ 2 3 dislocations, one planar and two non-planar. Li and Ma [8] reported first-order pyramidal slip, {1 0 1¯ 1} 1¯ 1¯ 2 3 in Mg. In simulation studies, pyramidal dislocations have been created in single crystals either by manually moving some atoms [9–15] or by applying external stress to an artificial cavity [8]. It is not possible to investigate the initiation of c + a dislocations when they are constructed by hand. In creating a c + a dislocation from a cavity, the only source of the pyramidal dislocation is the cavity itself. It is thus hard to see various activation processes of c + a slip that might be active in polycrystalline materials, in which grain boundaries (GBs) can be a source of dislocations [16].
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Fig. 1. (a) The first- and second-order pyramidal slip planes in an HCP unit cell. (b) Columnar microstructure with [1 1 2¯ 0] or [1 0 1¯ 0] texture.
This study focuses on the heterogeneous nucleation of c + a dislocations from GBs in polycrystalline Mg. In particular, the structure and properties of both first- and second-order pyramidal c + a dislocations in Mg are compared.
2. Simulation approach All of the nanocrystalline structures simulated here consist of four columnar grains of 40 nm diameter. Each grain in the structure has the same crystallographic orientation along the zaxis; thus the grains are separated by tilt grain boundaries. Two different orientations are considered, as shown in Fig. 1: one with [1 1 2¯ 0] texture, the other with [1 0 1¯ 0] texture. The [1 1 2¯ 0] and [1 0 1¯ 0] textures have their z-axes parallel to the first- and second-order pyramidal c + a slip planes, {1 0 1¯ 1} and {1 1 2¯ 2}, respectively. In these textures, a hexagonal unit cell has twofold symmetry along the z-direction of the simulation cell, with the crystallographic c-axis parallel to the x–y plane. The interatomic interactions are described by an embedded atom method (EAM) potential [17]. The deformation simulations are conducted at 293 K with 3d-periodic boundary conditions applied to the simulation cell. The time step is t ≈ 0.4 fs. A uniaxial tensile load is applied along the x-direction in creep mode: the magnitude of the applied stress is constant within each simulation, with values ranging from 0.9 to 1.45 GPa. The Parrinello–Rahman constant stress algorithm [18] and the Nose–Hoover thermostat [19] are also implemented. The activation of specific dislocations at specific values of applied stress is determined from these creep tests. Common neighbor analysis (CNA) [20,21] is used to distinguish atoms in FCC-like stacking faults from HCP coordinated atoms. The ‘AtomEye’ software is used for visualization of MD results [22]. Complete details of the simulation set up are described in an earlier paper [23].
3. Simulation results The [1 1 2¯ 0]- and [1 0 1¯ 0]-textured structures manifest different pyramidal dislocations under tensile stress. The fundamentals of the shape and nucleation condition of each pyramidal dislocation are compared in this section. In addition, dislocation structures are analyzed in detail.
3.1. Occurrence of pyramidal dislocations As reported in an earlier paper [23], pyramidal c + a slip processes are common in the [1 1 2¯ 0] Mg columnar texture. To activate pyramidal slip in this work, the [1 1 2¯ 0]-texture is strained under uniaxial stress of 1.18 GPa. After activation of pyramidal dislocations, the structure is quenched at 0.01 K. The structure is characterized with CNA to distinguish FCC from HCP environments. Gray, brown and black denote atoms in HCP environments, FCC environments (in stacking faults) and non-twelvefold coordinated, i.e., non-HCP or FCC (in dislocations and grain boundaries). In the [1 1 2¯ 0]-textured structure, only partial dislocations are generated from GBs under low applied stresses. As discussed previously [23], the structure labeled ˛ in Fig. 2(a) is a 1/62 0 2¯ 3 pyramidal c + a dislocation. At higher applied stresses, a transition of the partial c + a dislocation into two different structures takes place. Fig. 2(b) and (c) shows the structures for a stress of 1.45 GPa. One of the regions of the c + a partials is of the extended type, as seen in Fig. 2(b). The extended c + a pyramidal dislocations are denoted ˛ . The other high stress region is denoted ˛ in Fig. 2(c). It seems not to be an extended dislocation, but rather the result of the connection of long and short partial pyramidal dislocations. In these simulations, the ˛ structure is generally produced at higher stress than the ˛ . Pyramidal c + a slip is also found in a simulation of the [1 0 1¯ 0]-textured microstructure at an applied stress of 1.24 GPa. In Fig. 2(d), ˇ and ˇ indicate partial and extended pyramidal c + a dislocations, respectively. However, c + a slip nucleation appears to be less frequent in the [1 0 1¯ 0]-texture than in the [1 1 2¯ 0]-texture. Moreover, there is a difference between c + a partials in the [1 1 2¯ 0]- and [1 0 1¯ 0]-textured polycrystals. The c + a partials can glide through a grain in the [1 1 2¯ 0] texture, via a stacking fault mechanism. However, after being activated in the [1 0 1¯ 0]-texture, the partial denoted as ˇ in Fig. 3(d) is unstable, and hence immediately transforms into an extended c + a dislocation.
3.2. Structures of c + a dislocations We first examine the partial and extended dislocations appearing in the [1 1 2¯ 0]-textured structure. To understand the mechanism of partial slip, the structure of the leading core connected to a staking fault is illustrated in Fig. 3; the c + a dislocation is seen as black atoms among the atoms in HCP environments (gray). Blue dots denote the positions of the atoms before slip.
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Fig. 2. Snapshots of pyramidal c + a slip activated in [1 1 2¯ 0]- and [1 0 1¯ 0]-textured structures. Gray, black and brown denote normal (HCP), disordered (non-HCP or FCC), and stacking fault (FCC) atoms, respectively. Arrows denote the directions in which the c + a dislocations move. (a) The partial c + a pyramidal (˛) and basal dislocations, shown as brown lines, are nucleated at 1.18 GPa. (b) Extended pyramidal dislocation, ˛ , is activated at 1.45 GPa. Its shape changes during glide. (c) Two connected partial dislocations, ˛ , appear at 1.45 GPa. (d) Pyramidal c + a dislocations with partial (ˇ) and extended types (ˇ ) are found during deformation in [1 0 1¯ 0]-textured Mg. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The HCP stacking sequence in a basal direction is represented as alternating ‘A’ and ‘B’ layers. Green arrows denote directions of the shear stress that causes the slip. In the non-basal SF, the atomic displacement, indicated by the yellow arrows is parallel to [1 1¯ 0 2] and normal to [1 1 2¯ 0]. These two directions are on the {1 0 1¯ 1} plane between two c + a pyramidal slip planes previously illustrated at Fig. 1(a); hence, the depicted dislocation is a first-order pyramidal partial. The observed dislocation has a partial core and staking fault (SF), as shown in Fig. 3(b). The pyramidal c + a dislocation line exists in the [1 1 2¯ 0] direction, indicated as ‘⊥’ in Fig. 3(b). The potential energy (PE) of atoms in dislocation structure is also shown in Fig. 3(b). Although the atom positions show some disorder at the core of the partial dislocation, a stacking fault is formed by coordinated atomic motion. The two layers of the stacking fault colored in black move in opposite directions: one along 1/12[2 0 2¯ 3], ¯ These opposite atomic displacements the other along 1/12[2¯ 0 2 3]. ¯ partial slip, as shown previously combine to produce 1/6[2 0 2 3] [23]. An extended pyramidal dislocation in the [1 1 2¯ 0]-texture is more complicated than the partial dislocation. CNA results and PEs of the extended dislocation are shown in Fig. 4(a) and (b), respectively. The extended dislocation can be divided into leading and trailing partials connected with a SF, see Fig. 4(a). The regions with high PE in Fig. 4(b) correspond to the leading and trailing partial cores. Extra-half planes are found near the cores. There is a boundary of layers, denoted by the dotted (blue) line, at the upper side of the extended dislocation. The nature of this boundary will be dis-
cussed below in connection with sequential slip that leads to the formation of an extended dislocation. The structures of the leading and trailing partials are characterized in detail in Fig. 5. Among the three stacking fault lines (layers I, II, and III) of the trailing core, layers II and III move in opposite directions, as indicated by the yellow arrows in Fig. 5(a). To quantitatively determine their displacements, the atomic structure of layers I, II, and III are illustrated in Fig. 5(c). The slip of layers II ¯ They thus form a total of and III is 1/12[2 0 2¯ 3] and 1/12[2¯ 0 2 3]. 1/6[2 0 2¯ 3] partial slip, in the same manner as shown earlier in Fig. 3. The atomic structure of the dislocation core in the leading partial is also exhibited in Fig. 5(b). Each atom moves in the direction denoted by the yellow arrows. Some of atomic displacements are not parallel to the glide direction of the extended pyramidal dislocation in the leading core of Fig. 5(d). In the leading core, all the atomic displacements collectively make two components of partial slip: 1/18[4 2 6¯ 3] and 1/9[1 1¯ 0 3]. Fig. 6 shows the Burgers vectors of the extended pyramidal dislocation in the HCP unit cell. The three partials observed in the leading and trailing parts produce perfect pyramidal slip as follows: C + B0 + B0 B → CB 1/9[0 1¯ 1 3] + 1/18[6¯ 2 4 3] + 1/6[0 2¯ 2 3] → 1/3[1¯ 1¯ 2 3]
(1)
In the above, it has been demonstrated that the c + a dislocations in the [1 1 2¯ 0]-textured structure lie on the first-order
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Fig. 4. CNA and PE map of the extended pyramidal dislocation observed in the [1 1 2¯ 0]-textured structure. (a) ˛ denotes the extended c + a dislocation. Blue dots indicate a boundary of different layers: A and B denote different layer in the direction of [1 1 2¯ 0]. (b) The white ‘⊥’ represent the cores of the edge dislocations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 3. Common neighbor analysis (CNA) and potential energy (PE) map of the partial pyramidal dislocation in the [1 1 2¯ 0]-textured structure. The white ‘⊥’ denote the cores of the edge dislocations in (b).
pyramidal plane, appearing either as partials or as fully extended dislocations. Now we focus on the pyramidal dislocations found in the [1 0 1¯ 0]-texture. In order to characterize their structure, the atomic-level details of the c + a dislocation found in the [1 0 1¯ 0] texture of Fig. 2 are shown in Fig. 7. The CNA result in Fig. 7(a) shows that the dislocation has an extra half-plane parallel to the basal direction. The dislocation seems to be divided into two different parts, as seen in the potential energy map of Fig. 7(b). The head partial, which contains a series of high-energy atoms, lies parallel to the z-direction, [1 0 1¯ 0]. The left image of Fig. 7(b) shows that the head core spreads into the basal plane. This basal plane connects to the tail part, seen as an angular plane. This shape is quite different from the extended pyramidal dislocation formed in the [1 1 2¯ 0]-textured structure. Because the structures of the head and tail partials are keys to understanding a formation mechanism of
the extended c + a dislocation, they are displayed in further detail in Fig. 8. Fig. 8 shows CNA images projected on both the (1 0 1¯ 0) and (0 0 0 1) planes in the two left columns, while the potential energy (PE) on (0 0 0 1) is mapped in the right column. The firstand second-order pyramidal c + a slip planes are seen as semitransparent red and blue. Blue dots and yellow arrows denote the sites of atoms before slip and the atomic displacement, respectively. The stacking sequence of HCP in the basal direction is also seen as ‘A’ and ‘B’. Slip on the secondary c + a plane can be seen in the CNA results of Fig. 8(a). Referring to the (0 0 0 1) oriented CNA figure, the yellow arrows indicate that the atoms do not displace in the same direction as in secondary pyramidal slip. The difference in the slip behavior at each atomic layer parallel to the slip plane is due to the motion of zonal dislocations known to typically occur in pyramidal slip [1]. The zonal dislocations appear as zigzag ‘shuffling’ displacements. They are also known to make displacement of ¯ see Fig. 6 [1]. Most of the atomic displacements ∼1/4B C ([1 1 2¯ 3]), shown in the vicinity of secondary pyramidal slip plane, {1 1 2¯ 2}, of Fig. 8 are also ∼1/4B C. In addition, the PEs of atoms on the slip plane are higher than those of bulk HCP atoms. The next basal layer of the cross section of Fig. 8(a) along is shown in Fig. 8(b). The secondary pyramidal slip has progressed more in Fig. 8(b), as evidenced by atomic displacements and PEs on the slip plane of Fig. 8(b). As shown in the (0 0 0 1)-oriented CNA result of Fig. 8(a), the atomic displacement takes place on one layer parallel to the secondary c + a slip plane of (1 1 2¯ 2); yellow arrows form a monolayer in the [1 1 2¯ 0] microstructure. Two layers move, however, by individual displacement of atoms adjacent to the slip plane, as shown by the two yellow arrows in the (0 0 0 1)-oriented CNA result of Fig. 8(b).
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Fig. 5. Atomic structures of a (a) leading and (b) trailing of the extended c + a dislocation including CNA. (c) and (d) corresponding atomic displacement on the first-order pyramidal slip plane, (1 0 1¯ 1).
As the secondary pyramidal slip progress, a region of atoms with high PEs appears between the secondary pyramidal slip plane and an angular boundary in the (0 0 0 1)-oriented PE map of Fig. 8(b).
Fig. 6. Burgers vectors of the first-order pyramidal extended dislocation shown in the HCP unit cell.
This implies that additional displacements occur on a basal plane by the motions of zonal dislocations on a second pyramidal plane. The stress on the basal plane thus activates slip on the first-order pyramidal plane of (1 0 1¯ 1), as exhibited in Fig. 8(c). The extended pyramidal dislocation is thus revealed to contain both first- and second-order pyramidal slip. As already observed in Fig. 7(a), the secondary pyramidal slip is edge type, as evidenced by an extra half plane of atoms. However, because the first-order c + a partial slip does not show an extra-half plane, it appears to be a screw dislocation. Fig. 9 exhibits the tail partials of the extended c + a dislocation structure in the [1 0 1¯ 0] texture. The semi-transparent red and blue sheets denote the first- and second-order pyramidal slip planes, respectively. The atomic displacements associated with slip are represented by green arrows. The edge dislocation line of the secondary c + a slip is denoted te in Fig. 9(a) and (b). The first-order pyramidal slip is initiated at Fig. 9(b). As shown in the (0 0 0 1)oriented CNA result of Fig. 8(c), the atomic motions occur along 1 1 2¯ 0. However, as Fig. 9(c) shows, the atomic motions change to the pyramidal c + a direction. The presence of a screw dislocation can be seen in Fig. 9(g) and (h). Although the same plane is seen in both, in Fig. 9(g) line ˛ is disconnected, and line ˇ is continued, while in Fig. 9(h) the line ˛ is continued, and line ˇ is disconnected. This feature is typical of screw dislocations. Hence, the pyramidal dislocation at the trail partial is screw type, having a dislocation line, ts .
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Fig. 7. (a) CNA and (b) PE map of the pyramidal c + a dislocation activated in the [1 0 1¯ 0]-textured structure.
Fig. 8. CNA visualization and PE map of the extended c + a pyramidal dislocation projected onto the (1 0 1¯ 0) and (0 0 0 1) planes. Atomic structures where one and two layers of top basal surface are removed at (a) are shown at (b) and (c), respectively. ‘A’ and ‘B’ represent the stacking sequence of HCP in the basal direction. The shear direction is marked by the green arrow. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 9. Layer structures of the extended pyramidal c + a dislocation in the [1 0 1¯ 0]-texture by CNA. The semi-transparent red and blue sheets denote the first- and secondorder pyramidal slip planes, respectively. ‘A’ and ‘B’ represent the stacking sequence of HCP in the basal direction. The direction of local slip is seen as a green arrow. te and ts indicate a dislocation line of screw and edge, respectively. ˛˛ and ˇˇ denote discontinuous atomic layers in the basal direction. However, each atom has less level difference ˚ in continuous lines of ˛ or ˇ. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this than atomic radius (1.6 A) article.)
4. Discussion Comparing the [1 1 2¯ 0]- and [1 0 1¯ 0]-textured structures, there are some significant points to be addressed with regards to their c + a slip behaviors. In this section, the previously presented results are analyzed in terms of c + a dislocation structures. The differences in the formation and properties of first- and secondorder pyramidal dislocations are discussed.
1/6[2¯ 0 2 3] + 1/6[0 2¯ 2 3] → 1/3[1¯ 1¯ 2 3]
In the above analyses, both first-order [5,8,9,14,15] and secondorder [4,6,7,10–13] pyramidal dislocation structures have been considered. For the first-order pyramidal plane, Liang and Bacon [9] studied the pyramidal slip on the {1 0 1¯ 1} plane using a Lennard–Jones potential. They identified a dissociation reaction:
(5)
Morris et al. [13] suggested a Type III variant, in which the dissociation is different: 1/3[1 0 1¯ 0] + 1/3[0 1 1¯ 3] → 1/3[1¯ 1¯ 2 3]
4.1. Dislocation structures
1/4[1¯ 0 1 2] + 1/12[1 4¯ 3 2] + 1/6[1 0 1 2] → 1/3[1¯ 1¯ 2 3]
The secondary c + a edge dislocation is known to display three variants [11,13]. Type I is a perfect dislocation, while Type II consists of two 1/2c + a partials. The Type I transforms to the Type II on heating from 0 K to 293 K [7]. The dissociation reaction of Type II is:
(6)
Partial edge cores of Type II are found in the [1 0 1¯ 0]-textured structure in our simulations. In contrast to these reactions, pyramidal dislocations in the [1 0 1¯ 0]-textured structure have both edge and screw components. Considering a screw dislocation, there are two issues to be compared to previous studies. First, the screw dislocation is likely to cross-slip: it has been reported that a c + a
(2)
Li and Ma suggested a dissociation process of the first-order c + a dislocation in Mg: 1/6[2 1¯ 1 0] + 1/6[0 1 1¯ 3] + 1/6[0 2 2 3] → 1/3[1¯ 1¯ 2 3]
(3)
Although their Mg EAM potential [17] is the same as that used in this work, the reaction in Eq. (3) is quite different from that seen in our simulations: 1/9[1¯ 0 1 3] + 1/18[2 6¯ 4 3] + 1/6[2¯ 0 2 3] → 1/3[1¯ 1¯ 2 3]
(4)
This mechanism was previously reported by Jones and Hutchinson [5]. Using a hard sphere model for a titanium alloy (Ti–6Al–4V), they characterized the first-order pyramidal slip as dissociated. Thus, at this point, there is a question as to why the same dissociation process takes place in both Mg and Ti. While Mg has an essentially spatially isotropic electronic structure of s and p orbitals [23], the 3d2 orbitals of Ti make it strongly anisotropic [23]. Such an anisotropy is responsible for the non-ideality in the c/a ratio and differences in the dominant slip mode: Mg and Ti have basal and prismatic slip as a dominant slip mode, respectively [24]. It is important to note that the work by Jones and Hutchinson [5] was performed using an intrinsically isotropic hard-sphere model. Hence, it is not surprising that the partial dissociations they observed are the same as those we see in Mg.
Fig. 10. Potential energy distributions (snapshots) and strain energy curves within cylinders of radius R having a center of an edge pyramidal c + a dislocation line.
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Fig. 11. Pyramidal c + a dislocations simulated by Sun potential. (a) CNA image of [1 1 2¯ 0] texture at 1.45 GPa; black, brown and cyan denote disordered (non-HCP or FCC), stacking fault (FCC) atoms, and twinned area. ˛1 is connected with FCC stacking faults, and ˛2 shows two c + a dislocations joined on two different slip planes. (b) Potential energy analysis of [1 0 1¯ 0] texture at 1.32 GPa in (a). Arrow indicates the directions in which the c + a dislocations move ˛ and ˇ denote pyramidal c + a and basal a dislocations. ˛ has the same structure as the extended c + a dislocation illustrated in Fig. 7. An intergranular crack appears between grains 2 and 4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
screw dislocation spreads on two {1 0 1¯ 0} first-order pyramidal planes at 0 K [7]. Yoo et al. [3] concluded that a c + a screw dislocation may split on multiple slip planes, e.g. {1 0 1¯ 0} and {1 1 2¯ 2}. In the [1 0 1¯ 0]-textured structure the pyramidal c + a dislocation takes place by cross-slip on the first- and second-order slip planes mediating a basal plane (see Figs. 8 and 9). The screw core also appears as a junction of two {1 0 1¯ 0} planes in this simulation. Second, the secondary c + a edge dislocation becomes immobilized: the secondary pyramidal edge dislocations of Type II is known to be sessile at >30 K because of the extended core along the (0 0 0 1) [7]. Type III is also sessile with a basal stacking fault, as seen in Eq. (6) [13]. This spreading of the second-order pyramidal dislocation on the basal plane was already observed in Figs. 7–9. This non-planar core spreading in HCP causes dislocations to be sessile [24]. The immobilization of secondary c + a edge dislocations may be one of the reasons why it is difficult to form long partial c + a dislocations in the [1 0 1¯ 0]-texture, as previously mentioned in Section 3.1. 4.2. Activation of first- and second-order c + a slip The simulations described above show that c + a pyramidal slip processes are more easily observed in the [1 1 2¯ 0]-texture than in the [1 0 1¯ 0]-texture. Hence, first-order pyramidal slip in the [1 1 2¯ 0]-texture takes place more easily than second-order
pyramidal slip in the [1 0 1¯ 0]-texture. This is in disagreement with the experimental findings that secondary pyramidal slip on 1/3{1 1 2¯ 2} 1¯ 1¯ 2 3 dominates in Mg. To address this discrepancy, we compare the atomic motion, potential energies, and stacking fault energies of pyramidal edge dislocations in the two textures. In the [1 1 2¯ 0]-texture the first-order partial pyramidal dislocation, 1/6[2 0 2¯ 3], arises from partial motion of two adjacent layers, ¯ as previously shown in Fig. 3. ∼1/12[2 0 2¯ 3] and 1/12[2¯ 0 2 3], By contrast, the core of secondary pyramidal slip with edge type is obtained by full motion of only one layer with zonal behavior in Fig. 8. For the purposes of an energy analysis of the firstand second-order pyramidal dislocations, the distributions of total strain energy per Burgers vector are represented in Fig. 10 as a function of distance from the centers of dislocation cores. The total strain energy (Etotal ), expressed by adding core and elastic strain energy (Ecore + Eelastic ) [25], is useful for analyzing the size and energy of a dislocation core. Pyramidal dislocations are partial or extended types in this work: the 1st and 2nd pyramidal dislocations are connected with a {1 0 1¯ 1} SF and with screw dislocation consisting of two {1 0 1¯ 1} planes, respectively. In order to focus on the edge cores of pyramidal slip, we examine the total strain energy of core regions (Ecore ) rather than that of the elastic region (Eelastic ). The magnitude of Burgers vector (b) for both partial pyramidal slip ˚ the total strain energy is plotted to 3.75b, as shown in is 3.2 A;
Table 1 Stacking fault energies in magnesium. Stacking fault energy (Esf , mJ/m2 ) Basal a, {0 0 0 1} ISF-1
First-order c + a slip plane, {1 0 1¯ 1}
Method Second-order c + a slip plane, {1 1 2¯ 2}
ISF-2
This work
24
48
122
253
MD (poly), Liu EAM
Nogaret [29]
27 – –
54 44 –
121, 225 118, 240 180
198 × 236
MD (single), Liu EAM MD (single), Sun EAM Ab initio
Morris [30] Chetty [31] Jelinek [32] Yasi [33] Wang [34]
– – 18 –
– 44 37 34 21
– – – –
173 – – –
– – – –
78 60 >90 <50
– – – –
– – – –
Sastry [35] Devlin [36] Fleischer [37] Couret [38]
Experiment
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of basal intrinsic stacking fault-1 (ISF-1) and -2 (ISF-2) are less than those which were reported by Liu [17]. This discrepancy may be caused by imperfect relaxation of our polycrystalline structure due to a residual stress after partial slip process; by contrast single crystals having no other defects would allow complete stress relaxation. However, our SFEs are still in the range of values experimentally reported. As shown in Table 1, the SFE on the second-order c + a slip plane, 253 mJ/m2 is twice that of the SFE on the first-order c + a slip plane, 122 mJ/m2 . The second-order pyramidal c + a dislocations did not show long SFs connecting their partial cores, whereas long SFs are easily produced in the first-order c + a dislocations. Presumably, this is a result of the differences in their SFEs. In summary, the atomic motions and partial core energies explain the more frequent occurrence of the first-order pyramidal dislocation. Although the second-order c + a edge dislocations nucleate GBs, their non-planar structure prevent them from gliding inside a grain. As local stress is concentrated at these sessile dislocations, however, the nucleated partials become mobile. Due to their high SFEs, they normally appear as extended dislocations rather than partial cores separated by long SFs. Moreover, the high SFE of second pyramidal c + a slip indicates that the second-order c + a slip are likely to be a screw type or dissociated (c + a → sessile c + glissile a) processes rather than pure edge type, as reported in previous studies [4]. It is instructive to compare these simulation results with experimental findings. Tonda and Ando [6] reported a difference in yield shear stress between first- and second-order pyramidal slip of only ∼20%. This difference is small comparable to the critical resolves shear stress (CRSS) ratio for non-basal to basal slip. This indicates that first-order pyramidal slip of 1/3{1 0 1¯ 1} 1¯ 1¯ 2 3 should also be manifested for appropriate textures and applied stresses; activation of deformation modes in HCP metals, such as Mg, strongly depends on texture and applied stress [39]. Indeed, Li and Ma [8] also suggested that first-order pyramidal slip takes place in MD simulation of single crystalline Mg, and identified the associated stacking faults in TEM [40].
4.3. Comparison of different EAM potentials
Fig. 12. Creep curves and snapshots of strained columnar Mg with 18 nm grain size. Cyan, brown and black in (b) and (c) mean a twinned region, stacking fault (FCC), disordered (non-HCP or FCC), respectively. (a) Time–strain curves at 1.035 and 1.18 GPa. (b) Snapshot at ˛ of 1.18 Gpa-curve. (c) Snapshot at ˇ of 1.035 Gpacurve. A red circle indicates a region where an interaction between twinning and c + a dislocations take place. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 10. Taking the dislocation core size to be 3b [25–28], it is possible to estimate the core energy from Fig. 10. The second-order pyramidal dislocation core in the [1 0 1¯ 0]-texture is found to have much higher strain energy than the first-order pyramidal slip in the [1 1 2¯ 0]-texture. In addition, analysis of stacking faults energies (SFEs) in Mg from various simulation and experimental methods are compared in Table 1. The SFEs in the present work were determined from SF regions >5 nm apart from a core, created in polycrystalline by partial slip. Since there has been little analysis of SFEs on the c + a plane, we tried to obtain consistency of our results through comparing well-known SFEs on the basal plane. The calculated energies
The properties of a material as determined in a simulation can depend strongly on the description of the interatomic interactions. There are two literature EAM potentials for Mg: one developed by Liu et al. [17], the other developed by Sun et al. [41]. To this point, we have adopted the Liu potential to simulate plastic deformation of nanocrystalline Mg [23]. The EAM potential fitted by Sun et al. has been found to manifest more active basal slip behavior than the Liu potential [42], a result of the lower Peierls stress [29]. We have compared the deformation behavior as given by the Sun potential with the results above for the Liu potential. We find that there are far fewer nucleation events for twinning and c + a slip for the Sun potential [42]. In addition, intergranular cracks are frequently seen even at small plastic strains (<3%) during plastic deformation process of polycrystalline sample, as shown in Fig. 11(b), presumably a consequence of the reduced twinning and c + a slip activity. Such cracks are common in simulation of 3D polycrystalline Mg using Sun potential [42]. As described by the Liu potential, the first-order pyramidal c + a partial slip in the [1 1 2¯ 0] texture shows partial dislocations joined by long SFs; moreover the extended c + a dislocations glide easily. In contrast, Sun potential produces comparatively short non-basal SFs often connected with a SF of FCC on the basal plane, as exhibited in Fig. 11(a), and less mobile extended c + a dislocations. Comparing the structure of c + a dislocations using Liu and Sun potentials in [1 1 2¯ 0] and [1 0 1¯ 0] textures of Mg, there was no significant difference (Fig. 11), consistent with the results of [29].
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4.4. Role of pyramidal c + a slip in a plastic deformation The pyramidal c + a dislocations are found to play a number of roles. First, homogeneous plastic deformation requires pyramidal c + a slip [43,44]. A basal or prismatic mode appears as dominant slip in HCP metals. However, if some grains require high CRSS to activate the basal or prismatic mode, the secondary slip mode, pyramidal c + a, should be activated for plastic deformation. Activation of this secondary mode has a direct effect on mechanical response of the material at meso- and macro-levels. Single crystalline textures of Mg show noticeable differences in stress–strain curve according to the directions in which stress is applied [45]. The activation of secondary c + a slip is one of the important reasons for these differing deformation responses. Second, the interaction between c + a slip and twinning results in hardening. The twinning and c + a slip compete in grains in which basal slip is rare. Fig. 12 shows hardening from the interaction between the c + a dislocation and the twinning boundary. Initially the simulated [1 1 2¯ 0] textures are defect-free, except for the GBs. The interaction can be thus examined between dislocations and twinning created under the external stress. To maintain the specific stress value needed for the c + a slip–twinning interaction, a creep test is more suitable than a tensile test. The initial textures are strained at 1.035 and 1.18 GPa, showing a c + a dislocation–twinning interaction and only twinning in Fig. 12(b) and (c), respectively. While the 1.18 GPa-curve shows the strain rate (εA ) of 1.5 × 109 s−1 , a significant decrease (εB = 1.4 × 108 s−1 ) of stain rate appears in 1.035 GPa-curve of Fig. 12(a). As shown in Fig. 12 (b) and (c). We attribute this hardening to growth of the {1 0 1¯ 2} 1 0 1¯ 1 tensile twinning blocked by the c + a dislocations of order {1 0 1¯ 1} 1¯ 1¯ 2 3. In addition, it was experimentally reported that under compressive stress the c + a slip has a harmful effect on room temperature ductility of Mg: it produces severe strain hardening by dissociation to mobile a and immobile c dislocations [4,46]. 5. Conclusions Pyramidal c + a dislocations were created at GBs in [1 1 2¯ 0]and [1 0 1¯ 0]-textured Mg using MD simulation. It was found that both the first- and second-order pyramidal slip can be activated in Mg. The first-order c + a dislocations in the [1 1 2¯ 0]-texture manifested various forms, depending on the external stress. At low stress they are 1/62¯ 0 2 3 partials. Two phases, extended dislocation and connected partials, appear at high stress. The extended type of the first-order pyramidal slip takes place through a dissociation reaction of 1/9[0 1¯ 1 3] + 1/18[6¯ 2 4 3] + 1/6[0 2¯ 2 3] → 1/3[1¯ 1¯ 2 3]. While all of the dislocations in the [1 1 2¯ 0] texture are of edge type, the extended secondary pyramidal dislocation in the [1 0 1¯ 0] texture exhibits both types of edge and screw. The edge partial appears as a leading core of the secondary c + a slip, formed by typical zonal dislocation in HCP metals. The partial core, due to spreading on the basal plane and high SFE, remain sessile. During deformation, stress concentration which occurs at the sessile dislocation finally causes an activation of screw dislocation on the basal plane connected with the edge core. Hence, the complex dislocation is possible to glide in the [1 0 1¯ 0] columnar textures without leaving SF having high energy. Comparing the two different EAM potentials, the atomistic structures of pyramidal c + a dislocations are found to be very similar, though the Liu potential shows more active c + a slip than Sun potential. The textured columnar nano-structures considered here are in many ways ideal for fundamental studies; however, they are opti-
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