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twinning and
detwinning in magnesium
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 015003 (10pp)
doi:10.1088/0953-8984/26/1/015003
Atomic simulations of (101¯ 2), (101¯ 1) twinning and (101¯ 2) detwinning in magnesium Motohiro Yuasa1 , Makoto Hayashi2 , Mamoru Mabuchi2 and Yasumasa Chino1 1
Materials Research Institute for Sustainable Development, National Institute of Advanced Industrial Science and Technology, 2266-98 Anagahora, Shimo-shidami, Moriyama, Nagoya 463-8560, Japan 2 Graduate School of Energy Science, Kyoto University, Yoshidahonmachi, Sakyo, Kyoto 606-8501, Japan E-mail: [email protected] Received 8 August 2013, in final form 8 October 2013 Published 22 November 2013 Abstract
¯ and (1011) ¯ twinning and (1012) ¯ detwinning for Mg were investigated from the (1012) viewpoint of mobility of twinning dislocations and atomic shuffling. First-principles ¯ calculations suggested that the twinning dislocations glide more readily for the (1011) ¯ twinning than for the (1012) twinning. However, this conflicts with the experimental fact of ¯ twin formation. On the other hand, molecular dynamics simulations showed that easier (1012) ¯ twinning than for the (1011) ¯ twinning, the atomic shuffling was more activated for the (1012) which corresponds to the experimental fact. Therefore, it is suggested that the rate-controlling process for the twin formation is the atomic shuffling. Moreover, the calculations and ¯ simulations showed that the twinning dislocations glide more readily for the (1012) ¯ twinning, whereas the atomic shuffling is less activated for the detwinning than for the (1012) detwinning, suggesting that the detwinning occurs easily but is unstable, resulting in easy repetition of twinning–detwinning. (Some figures may appear in colour only in the online journal)
1. Introduction
formed at an initial stage of deformation or in a small strain range, and it increases the uniform elongation [5] and the ¯ twin strain hardening [5–8]. On the other hand, the (1011) occurs at a final stage of deformation or in a large strain range, and it is often related to premature fracture [4, 9]. It is known ¯ that the critical resolved shear stress (CRSS) for the (1012) ¯ ¯ twin is lower than that for the (1011) twin and the (1012) twin ¯ twin [10], although is formed more readily than the (1011) ¯ the boundary energy of the (1012) twin is higher than that ¯ twin [11]. Thus, the degree of difficulty for of the (1011) twin formation in Mg does not depend on the twin-boundary energy. Consequently, one of the problems of deformation twinning in Mg is the rate-controlling process of the twin formation. Recently, the twin nucleation and growth mechanisms in Mg have been investigated by atomic simulations [12–23]. Wang et al [12, 13] showed that a stable nucleus of
Mg alloys are attracting much attention as light structural materials because they exhibit high specific strength and high specific stiffness. However, Mg alloys often exhibit low ductility and low plastic formability as a result of premature fracture [1, 2]. Hence, it is necessary to improve the poor ductility of these alloys in order to increase their applications. Deformation twinning is one of the important deformation modes for plastic deformation in hexagonal close-packed (hcp) metals such as Mg [3], and it strongly affects the mechanical properties. For example, deformation twinning is related to premature fracture [4]. Therefore, it is worthwhile to understand the characteristics of deformation twinning of Mg in order to improve its mechanical properties. The main modes of deformation twins in Mg are ¯ ¯ and (1011)[10 ¯ ¯ twins. The (1012) ¯ twin is (1012)[10 11] 12] 0953-8984/14/015003+10$33.00
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c 2014 IOP Publishing Ltd Printed in the UK
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¯ twin is created by glide of zonal dislocations the (1012) consisting of a partial dislocation and multiple twinning dislocations. The zonal twinning mechanism is energetically favorable relative to the normal twinning mechanism for ¯ the (1012) twinning [13]. Barrett et al [14] suggested that dissociation of matrix dislocations is required for the twin nucleation. Capolungo and Beyerlein [15] proposed a three-dimensional twin nucleation model based on the nonplanar dissociation of a basal partial dislocation by using the continuous linear elastic dislocation theory. Li and ¯ twinning mediated by zonal Ma [16] investigated the (1011) dislocations, which are related to the incomplete dislocation on the twinning plane. Wang et al [17] performed molecular ¯ twinning, and dynamics (MD) simulations of the (1011) they showed that a two-layer twinning dislocation, which is nucleated by the interaction of a basal dislocation with a twin boundary, induces the twin nucleation. These works pointed out the importance of the interactions of twin boundaries with matrix dislocations for the twin nucleation. ¯ However, Capolungo et al [18] noted that the (1012) twin growth is independent of the interaction. Moreover, multiscale modeling studies [19–21] have suggested that the twin nucleation most likely occurs at grain boundaries. Most recently, Xu et al [22] revealed the important role ¯ twinning. Thus, of prismatic/basal interfaces in the (1012) twin formation mechanisms are still in debate. These previous works have investigated the twin nucleation and growth mechanisms through glide of twinning (or zonal) dislocations. However, for hcp metals, the twin formation is not completed just by glide of twinning dislocations but also by atomic movement called atomic shuffling, which is required to complete the twin formation [24]. Zhang et al [25] showed by an experimental investigation with transmission ¯ twin boundaries deviated electron microscopy that the (1012) significantly from the twin plane. This fact suggests that both the atomic shuffling and the glide of twinning dislocations are critical for twinning. Hence, it is important to investigate the twin formation mechanisms in terms of both the glide of twinning dislocations and the atomic shuffling. Another problem related to twinning of Mg is ¯ (1012) detwinning. Many experimental works [26–32] ¯ have demonstrated that the (1012) detwinning occurs readily. For example, Wang et al [27] investigated the effects of compressive predeformation on subsequent tensile deformation of Mg with a ring fiber texture, and they ¯ twinning occurs under compressive found that the (1012) loading and the detwinning occurs during subsequent tensile reloading. Zhang et al [30] showed that twinning–detwinning induced the apparent Bauschinger effect. Hong et al [31] showed that fatigue resistance was improved by tailoring the twinning–detwinning characteristics. Thus, not only the ¯ plane occurs twinning, but also the detwinning of the (1012) readily and it affects the mechanical properties of Mg. If the atomic shuffling is critical for the twinning, it is expected to be critical for the detwinning as well, because the detwinning is not completed just by the glide of twinning dislocations. An aim of the present work is to investigate the mobility of twinning dislocations and the activity of atomic
shuffling in Mg by atomic simulations. Actually, the glide of twinning dislocation and the atomic shuffling may occur simultaneously for the twin formation. In the present work, however, atomic simulations are performed on the assumption that the twinning and the detwinning are complete by the atomic shuffling after the glide of twinning dislocations to investigate the rate-controlling process of the twinning and the detwinning. First-principles calculations of the generalized ¯ stacking fault energy (GSFE) are carried out for the (1012) ¯ twin planes to investigate the twin formation and the (1011) from the viewpoint of the mobility of twinning dislocations. ¯ and (1011) ¯ twinning Then, the atomic shuffling in the (1012) is investigated by relaxing the uncompleted twins after the glide of twinning dislocations using MD simulations, and ¯ twinning is the activity of atomic shuffling for the (1012) ¯ twinning. Moreover, the compared with that for the (1011) mobility of twinning dislocations and the atomic shuffling for ¯ detwinning are compared with those for the (1012) ¯ the (1012) twinning. 2. Calculation methods 2.1. Molecular dynamics simulation
¯ MD simulations were performed on the formation of (1012) ¯ twins through the glide of dislocations and the and (1011) atomic shuffling. Cell models used for the MD simulations are shown in figure 1, where the green circles indicate Mg atoms. Figure 1(a) shows a cell model prior to the glide ¯ twinning, where the x-axis is of dislocations for (1012) ¯ parallel to [1210], the y-axis is parallel to the twinning shear ¯ ¯ direction [1011], and the z-axis is normal to the (1012) plane. Figure 1(b) shows a cell model prior to the glide of ¯ twinning, where the x-axis is parallel dislocations for (1011) ¯ to [1210], the y-axis is parallel to the twinning shear direction ¯ ¯ plane. The [1012], and the z-axis is normal to the (1011) ¯ (1012) unit cell was 1.284 nm long (x-direction), 3.047 nm wide (y-direction), and 6.462 nm high (z-direction), and the ¯ unit cell was 1.284 nm long (x-direction), 2.362 nm (1011) wide (y-direction), and 8.827 nm high (z-direction). Periodic boundaries were adopted along the x- and y-axes. The analytic modified embedded atom method (AMEAM) potential was used. This potential is fit to experimental and ab initio data and reproduces many properties of Mg [33, 34]. The NVT ensemble with a cut-off distance of 0.642 nm was adopted. The integration time step was 1.0 fs throughout the MD simulations. The configurations of atoms were determined by geometry relaxation for 5 ps at 1 K. The temperature was controlled using the Nose–Hoover thermostat technique [35]. The relaxation conditions enabled the stabilization of the ¯ twinning was simulated configuration of atoms. The (1012) based on the zonal twinning mechanism, where the twin 57 ¯ [1011] partial dislocation and one nucleus consisted of one 107 1 ¯ 15 [1011] twinning dislocation [12, 13]. Concretely, the glide of dislocations was imported by the following operations: the atoms of layer 1 and above were displaced along the ¯ [1011] direction over a distance of 0.404 nm, resulting from the glide of a partial dislocation, and then the atoms of 2
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¯ ¯ twinning and (b) (1011) ¯ twinning. The Figure 1. [1210] projection of cell models used for the molecular dynamics simulations, (a) (1012) ¯ ¯ green circles indicate Mg atoms. In (a), the x-axis is parallel to [1210], the y-axis is parallel to the twinning shear direction [1011] and the ¯ plane. In (b), the x-axis is parallel to [1210], ¯ ¯ z-axis is normal to the (1012) the y-axis is parallel to the twinning shear direction [1012] and ¯ plane. the z-axis is normal to the (1011)
¯ layer 3 and above were displaced along the [1011] direction over a distance of 0.049 nm, resulting from the glide of ¯ twinning, first we a twinning dislocation. For the (1011) simulated the importing of twinning dislocation gliding on the ¯ plane based on the zonal twinning mechanism [16]. (1011) However, the twin was not formed through the glide of twinning dislocations and the atomic shuffling. Then, we simulated the glide of twinning dislocations based on the normal twinning mechanism. Namely, the atoms of layer 1 ¯ and above were displaced along the [1012] direction over a distance of 0.260 nm, resulting from the glide of a partial dislocation, and then the atoms of layer 2 and above were ¯ displaced along the [1012] direction over a distance of 0.260 nm, resulting from the glide of a twinning dislocation. ¯ twin As a result of the atomic shuffling after the glide, (1011) formation was completed, as shown later. After importing the dislocations, geometry relaxation was performed at 150–450 K. The atomic shuffling occurred during the relaxation. Figures 2(a) and (b) show the atomic configurations after the glide of dislocations and after the ¯ twinning, and atomic shuffling, respectively, for the (1012) figures 3(a) and (b) show the atomic configurations after the glide of dislocations and after the atomic shuffling,
¯ twinning. Twin formation was respectively, for the (1011) not completed just by the glide of dislocations; the atomic shuffling was also required for the stable structure of the twin. Consequently, twin formation was completed by the atomic ¯ and (1011) ¯ twinning. shuffling for both the (1012) In the present work, the diffusion coefficient was investigated, assuming that the atomic shuffling could be simulated by diffusion. The diffusion coefficient can be calculated by measuring the mean-square displacement (MSD) [36, 37]. According to the Einstein equation for three-dimensional Brownian motion, the diffusion coefficient can be given by D = hλ2 i/6t,
(1)
where D is the diffusion coefficient, hλ2 i is the MSD, and t is the time. The MSD, hλ2 i, can be given by hλ2 i = |1x2 + 1y2 + 1z2 |,
(2)
where 1x is the MSD along the x-axis, 1y is the MSD along the y-axis, and 1z is the MSD along the z-axis. The diffusion 3
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shuffling was investigated in the same manner as the diffusion coefficient. ¯ detwinning was simulated by the reverse The (1012) ¯ twinning. The atoms of layer 3 procedure to the (1012) ¯ direction over a and above were displaced along the [101¯ 1] distance of 0.049 nm (opposite to the twinning dislocation), and then the atoms of layer 1 and above were displaced along ¯ direction over a distance of 0.404 nm (opposite the [101¯ 1] to the partial dislocation). After these procedures, relaxation was performed to induce atomic shuffling at 150–450 K. The diffusion coefficient and activation energy of the atomic ¯ detwinning were calculated in the shuffling for the (1012) ¯ twinning. same manner as for the (1012) 2.2. First-principles calculations
To investigate the mobility of twinning dislocations, firstprinciples calculations were carried out in order to calculate ¯ and (1011) ¯ twin planes. The firstthe GSFE for the (1012) principles calculations were performed using the CASTEP code [38], in which the density functional theory [39, 40] was used with a plane wave basis set to calculate the electronic properties. The exchange–correlation interactions were treated using the generalized gradient approximation within the Perdew–Wang scheme (PW91). The energy cut-off was 350 eV. The ultrasoft pseudopotentials [41] represented in reciprocal space were used in the calculations. The Brillouin zone was sampled using the Monkhorst–Pack 4 × 4 × 1 ¯ and (1011) ¯ planes model, k-point mesh for both the (1012) and the Gaussian smearing with a width of 0.1 eV. The cell models used for the first-principles calculations of GSFE are shown in figure 4. The periodic boundary conditions were adopted along the x-, y-, and z-axes in ¯ both models. Figure 4(a) shows a cell model for the (1012) ¯ twinning, where the x-axis is parallel to [1210], the y-axis ¯ is parallel to the twinning shear direction [1011], and the ¯ ¯ z-axis is normal to the (1012) plane. The (1012) unit cell had periodic lengths of 0.762 nm along the x-axis, 0.642 nm along the y-axis, and 4.60 nm along the z-axis, and the number of the Mg atoms in the cell was 64. A vacuum spacing of 1.5 nm was made along the z-axis. Figure 4(b) is a cell model for ¯ twinning, where the x-axis is parallel to [1210], ¯ the (1011) ¯ the y-axis is parallel to the twinning shear direction [1012], ¯ plane. The (1011) ¯ and the z-axis is normal to the (1011) unit cell had periodic lengths of 1.181 nm along the x-axis, 0.642 nm along the y-axis, and 3.75 nm along the z-axis, and the number of Mg atoms in the cell was 80. A vacuum spacing of 2 nm was made along the z-axis. The GSFE was calculated by displacing one part of the crystal with respect to ¯ ¯ plane and the other along the [1011] direction on the (1012) ¯ ¯ along the [1012] direction on the (1011) plane. The orange and the blue circles in the upper part indicate shifted Mg atoms and the green circles in the lower part indicate non-shifted Mg ¯ twinning, the GSFE was calculated by atoms. For the (1012) ¯ displacing layers 1–9 (orange and blue circles) along [1011] over a distance of 0.404 nm. Then, the GSFE was evaluated ¯ by displacing layers 3–9 (blue circles) along [1011] over a ¯ distance of 0.049 nm. For the (1011) twinning, the GSFE was
¯ twinning, (a) after glide Figure 2. Atomic configurations for (1012) of twinning dislocations and (b) after atomic shuffling.
¯ twinning, (a) after glide Figure 3. Atomic configurations for (1011) of twinning dislocations and (b) after atomic shuffling.
coefficient can be given in another way by QA D = D0 exp − , kB T
(3)
where D0 is the pre-exponential factor, QA is the activation energy, kB is the Boltzmann constant, and T is the absolute temperature. In the present work, the activity of atomic 4
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¯ ¯ twinning and (b) (1011) ¯ twinning. The Figure 4. [1210] projection of cell models used for the first-principles calculations, (a) (1012) orange and the blue circles in the upper part indicate shifted Mg atoms and the green circles in the lower part indicate non-shifted Mg ¯ twinning is calculated by displacing layers 1–9 (orange and blue atoms. The generalized stacking fault energy (GSFE) for the (1012) ¯ ¯ ¯ twinning is evaluated by circles) along [1011], and then by displacing layers 3–9 (blue circles) along [1011]. The GSFE for the (1011) ¯ displacing layers 1–9 (orange and blue circles) along [1012], and then by displacing layers 2–9 (blue circles).
was 3.7 GPa for the glide of a partial dislocation and 5.3 GPa ¯ plane, for the glide of a twinning dislocation for the (1012) and 3.1 GPa for the glide of a partial dislocation and 3.0 GPa ¯ plane. for the glide of a twinning dislocation for the (1011) ¯ It is noted that the maximum slopes for the (1011) plane are ¯ plane. This indicates that the lower than those for the (1012) ¯ plane twinning dislocations glide more easily on the (1011) ¯ than on the (1012) plane. An inspection of figure 5(a) shows that the Burgers vector of the partial dislocation in the present 57 ¯ work is smaller than that by Wang et al [12], that is, 107 [1011], although the reason is unknown. The unstable stacking fault energy in the work by Wang et al is larger than the one in the present work. This supports the idea that the dislocation ¯ plane than on the (1012) ¯ glides more easily on the (1011) ¯ plane. However, it is known experimentally that the (1012) ¯ twin. This cannot twin is formed more easily than the (1011) be explained from the viewpoint of the mobility of twinning dislocations. The movement of atoms from their locations before relaxation to their locations after relaxation at 150 K by the MD simulation is shown in figure 6, where arrows indicate the movement of each atom before and after relaxation. It
evaluated by displacing layers 1–9 (orange and blue circles) ¯ along [1012] over a distance of 0.260 nm. It was then similarly evaluated by displacing layers 2–9 (blue circles). ¯ detwinning, the GSFE was In the case of the (1012) ¯ twinning calculated by the reverse procedure to the (1012) calculations. The GSFE was calculated by displacing layers ¯ direction over a distance of 0.049 nm, 3–9 along the [101¯ 1] ¯ direction over and then displacing layers 1–9 along the [101¯ 1] a distance of 0.404 nm. 3. Results and discussion
¯ ) and (1011 ¯ ) twinning 3.1. (1012 Figure 5 shows the variation in GSFE as a function ¯ and (1011) ¯ planes by the of the displacement in (1012) first-principles calculations. The unstable stacking fault energy and the unstable twinning energy were 25.6 and ¯ plane, and 19.3 ˚ −2 , respectively, for the (1012) 32.5 meV A −2 ¯ plane. Also, ˚ and 32.7 meV A , respectively, for the (1011) the maximum slope [42], characterized by the Peierls stress, 5
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¯ plane and (b) (1011) ¯ plane. The Figure 5. Variation in generalized stacking fault energy as a function of the displacement, (a) (1012) ˚ unstable stacking fault energy and the unstable twinning energy are 25.6 and 32.5 meV A ¯ plane, respectively. ˚ −2 for the (1011) 32.7 meV A
−2
¯ plane, and 19.3 and for the (1012)
Figure 6. Movement of atoms from location of atoms before relaxation to their location after relaxation at 150 K by MD simulation:
¯ plane and (b) (1011) ¯ plane. An arrow indicates the movement of each atom before and after relaxation. (a) (1012)
respectively. The energy of the cell decreased with increasing time in the short-time region less than 2–3 × 10−13 s, reaching a constant at the minimum value in the long-time region. Thus, the atomic movement behavior during relaxation was divided into two regions: the short-time region, where atomic movement was more active, and the long-time region, where atomic movement was less active. In figure 9, the 70% energy reduction point, i.e., the point at which the energy of the cell was reduced by 70%, ¯ twinning and at was reached at 1.0 × 10−13 s for the (1012) −13 ¯ 2.0 × 10 s for the (1011) twinning. The activity of atomic
can be seen that atoms move in various directions during the relaxation, resulting in complete formation of the twin. Similar behavior was obtained under relaxation at 200–450 K. The variations in MSD during relaxation at 150–450 K ¯ and the (1011) ¯ as a function of the time for the (1012) twinning are shown in figures 7 and 8, respectively. The atomic movement tended to be more activated in a short-time region of relaxation, less than approximately 2 × 10−13 s, than in a long-time region. Figures 9(a) and (b) show the variation in energy of the cell as a function of the time ¯ and (1011) ¯ twinning, during relaxation at 1 K for the (1012) 6
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Figure 7. Variations in mean-square displacement during relaxation
Figure 8. Variations in mean-square displacement during relaxation
¯ twinning. at 150–450 K as a function of the time for (1012)
¯ twinning. at 150–450 K as a function of the time for (1011)
¯ twinning, whereas minor differences were seen in the (1011) ¯ and the (1011) ¯ activity of atomic shuffling between the (1012) twinning in the long-time region. In typical atomic diffusion, the diffusion coefficient does not depend on the time. Hence, in the previous works [36, 37], long-time simulations were carried out in order to calculate the diffusion coefficient. In the present work, however, the activity of atomic shuffling depended on the time. This is because the atomic shuffling during relaxation is a result of atomic shuffling required for complete formation of the twin, thermal oscillation of atoms, and migration of the twin boundary. Therefore, it is suggested that the atomic shuffling in the long-time region is related to thermal oscillation of atoms and migration of the twin boundary after complete formation of the twin, and the
shuffling in the short-time region at each temperature was ¯ twinning and at 2.0× calculated at 1.0×10−13 s for the (1012) −13 ¯ 10 s for the (1011) twinning. Because the time needed to reach the 70% energy reduction point was determined at 1 K, the activity of atomic shuffling for complete twin formation at 150–450 K was necessarily in the short-time region. The variations in activity of atomic shuffling in the short- and long-time regions as a function of the inverse temperature for ¯ and (1011) ¯ twinning are shown in figures 10 and the (1012) 11, respectively. In figure 11, the activity of atomic shuffling in the long-time region was calculated at 1.0 × 10−12 s for ¯ twinning and the (1011) ¯ twinning. It is noted both the (1012) that the activity of atomic shuffling in the short-time region ¯ twinning is much larger than that for the for the (1012)
¯ twinning, (b) (1011) ¯ twinning, and Figure 9. Variation in energy of the cell during relaxation at 1 K as a function of the time, (a) (1012) ¯ detwinning. (c) (1012) 7
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Figure 11. Variations in activity of atomic shuffling (D) in the
long-time region as a function of the inverse temperature (1/T) for ¯ and (1011) ¯ twinning. (1012)
Figure 10. Variations in activity of atomic shuffling (D) in the
short-time region as a function of the inverse temperature (1/T) for ¯ and (1011) ¯ twinning. (1012)
¯ ) detwinning 3.2. (1012 atomic shuffling in the short-time region is mainly related to the atomic shuffling required for complete formation of the twin. As shown in figures 7 and 8, the MSD did not reach saturation after complete formation of the twin. This is because migration of the twin boundary occurred after complete formation of the twin. For improved reliability of the simulation, the activity of atomic shuffling was also calculated at times corresponding to the 50% and 90% energy reduction points, and similar trends were obtained. The value of QA for atomic shuffling in the short-time ¯ region was 5.9 meV/atom for the (1012) twinning and ¯ 18.1 meV/atom for the (1011) twinning. O’Handley et al [43] investigated twin-boundary motion in Ni–Mn–Ga and they showed that the activation energy for the twin-boundary motion was 15–30 meV/atom, which was in agreement with the experimental results of the Ni alloy. The activation energies in the short-time region in the present work are comparable to those found by O’Handley et al [43]. The ¯ atomic shuffling was more pronounced for the (1012) ¯ twinning than for the (1011) twinning at all temperatures and the activation energy for atomic shuffling was lower ¯ twinning than for the (1011) ¯ twinning in the for the (1012) ¯ short-time region. Clearly, the atomic shuffling in the (1012) ¯ twinning occurs more easily than in the (1011) twinning. ¯ This corresponds to the experimental fact that the (1012) ¯ twin is formed more easily than the (1011) twin [10]. Thus, the degree of difficulty of twin formation can be explained from the viewpoint of the atomic shuffling. Therefore, it is suggested that the rate-controlling process for complete formation of the twin is the atomic shuffling.
Figure 12 shows the variation in GSFE as a function of the ¯ detwinning. The GSFE increased displacement for the (1012) monotonically as a function of the displacement and the ˚ −2 in displacing layers 3–8 maximum GSFE was 5.5 meV A ¯ where the maximum slope was 3.1 GPa. The along [101¯ 1], GSFE continued to increase until reaching a displacement of 0.23 nm and then it decreased, and consequently the ˚ −2 in displacing layers 1–8 maximum GSFE was 30.7 meV A ¯ It is of interest to note that the maximum slope along [101¯ 1]. ¯ for the detwinning was in displacing layers 3–8 along [101¯ 1] ¯ for the lower than that in displacing layers 3–8 along [101¯ 1] twinning. The atomic shuffling after the glide of dislocations for the detwinning was investigated in the same manner as for the twinning. Because the time at which the 70% energy ¯ reduction point was reached was 1.7 × 10−13 s for (1012) detwinning (figure 9(c)), the activity of atomic shuffling in the ¯ short-time region was calculated at 1.7×10−13 s for the (1012) detwinning. The variation in activity of atomic shuffling as a ¯ detwinning function of the inverse temperature for the (1012) is shown in figure 13. The activation energy for atomic ¯ detwinning shuffling in the short-time region for the (1012) ¯ was 13.4 meV/atom, which is larger than that for the (1012) twinning, and also the activity of atomic shuffling in the short-time region for the detwinning was lower than that for the twinning. Hence, the atomic shuffling was less activated ¯ detwinning than for the (1012) ¯ twinning. for the (1012) The experiments showed that twinning–detwinning behavior occurs repeatedly by cyclic loading [28, 29, 31] and 8
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Figure 12. Variation in generalized stacking fault energy as a
¯ detwinning. function of the displacement for (1012)
that the CRSS for the detwinning is lower than that for the twinning [28, 30]. The present work revealed that twinning dislocations glide more easily for the detwinning, whereas the atomic shuffling is less activated, as compared to the twinning. Therefore, it is suggested that the detwinning occurs readily but it is unstable, resulting in a lower CRSS for the detwinning and the easy repeat of twinning–detwinning.
Figure 13. Variation in activity of atomic shuffling (D) in the
short-time region as a function of the inverse temperature (1/T) for ¯ detwinning. (1012)
4. Conclusions
¯ detwinning than for the (1012) ¯ twinning, whereas (1012) the atomic shuffling was less activated for the detwinning. Therefore, it is suggested that the detwinning occurs easily but it is unstable, resulting in a lower CRSS for the detwinning and easy repeat of twinning–detwinning.
¯ and (1011) ¯ twinning and (1012) ¯ detwinning for Mg (1012) were investigated in terms of the mobility of twinning dislocations and the atomic shuffling. First-principles calculations of the GSFE were carried out in order to investigate the mobility of twinning dislocations. Moreover, MD simulations were performed in order to investigate the atomic shuffling. The results are summarized as follows.
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(1) Twin formation was accomplished not only by the glide of twinning dislocations but also by the atomic shuffling for ¯ and the (1011) ¯ twinning. both the (1012) (2) The GSFE and its maximum slope for the glide of ¯ twinning twinning dislocations were larger for the (1012) ¯ than for the (1011) twinning. This conflicts with the ¯ twin is formed more experimental fact that the (1012) ¯ easily than the (1011) twin. Thus, the degree of difficulty for twin formation could not be explained from the viewpoint of the mobility of twinning dislocations. ¯ (3) The activity of atomic shuffling was larger for the (1012) ¯ twinning and the activation twinning than for the (1011) ¯ energy for atomic shuffling was lower for the (1012) ¯ twinning than for the (1011) twinning. Therefore, it is suggested that the rate-controlling process for twin formation is the atomic shuffling after the glide of twinning dislocations. (4) Also, the calculations and simulations showed that the twinning dislocations tended to glide more readily for the 9
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[31] [32] [33] [34]
[18] Capolungo L, Beyerlein I J and Tome C N 2009 Scr. Mater. 60 32 [19] Wang J, Beyerlein I J and Tome C N 2010 Scr. Mater. 63 741 [20] Beyerlein I J, Capolungo L, Marshall P E, McCabe R J and Tome C N 2010 Phil. Mag. 90 2161 [21] Beyerlein I J, McCabe R J and Tome C N 2011 J. Mech. Phys. Solids 59 988 [22] Xu B, Capolungo L and Rodney D 2013 Scr. Mater. 68 901 [23] Beyerlein I J, Wang J, Barnett M R and Tome C N 2012 Proc. R. Soc. Lond. A 468 1496 [24] Christian J W and Mahajan S 1995 Prog. Mater. Sci. 39 1 [25] Zhang X Y, Li B, Wu X L, Zhu Y T, Ma Q, Liu Q, Wang P T and Horstemeyer M F 2012 Scr. Mater. 67 862 [26] Kleiner S and Uggowitzer P J 2004 Mater. Sci. Eng. A 379 258 [27] Wang Y N and Huang J C 2007 Acta Mater. 55 897 [28] Lou X Y, Li M, Boger R K, Agnew S R and Wagoner R H 2007 Int. J. Plast. 23 44 [29] Wu L, Jain A, Brown D W, Stoica G M, Agnew S R, Clausen B, Fielden D E and Liaw P K 2008 Acta Mater. 56 688 [30] Zhang F, Hao M, Wang F C, Tan C W, Yu X D, Ma H L and Cai H N 2012 Scr. Mater. 67 951
[35] [36] [37] [38] [39] [40] [41] [42] [43]
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Hong S G, Park S H and Lee C S 2010 J. Mater. Res. 25 784 Sarker D and Chen D L 2012 Scr. Mater. 67 165 Zhang B, Ouyang Y, Liao S and Jin Z 1999 Physica B 262 218 Hu W, Xu H, Shu X, Yuan X, Gao B and Zhang B 2000 J. Phys. D: Appl. Phys. 33 711 Melchionna S, Ciccotti G and Hoover H B L 1993 Mol. Phys. 78 533 Chamati H, Papanicolaou N I, Mishin Y and Papaconstantopoulos D A 2006 Surf. Sci. 600 1793 Chen M, Hou Q, Wang J, Sun T, Long X and Lou S 2008 Solid State Commun. 148 178 Payne M C, Teter M P, Allan D C, Arias T A and Joannopoulos J D 1992 Rev. Mod. Phys. 64 1045 Hohenberg P and Kohn W 1964 Phys. Rev. B 864 136 Kohn W and Sham L 1965 Phys. Rev. A 1133 140 Vanderbilt D 1990 Phys. Rev. B 41 7892 Wang J, Beyerlein I J, Hirth J P and Tome C N 2011 Acta Mater. 59 3990 O’Handley R C, Paul D I, Allen S M, Richard M, Feuchtwanger J, Peterson B, Techapieasancharoenkij M, Barandiar´an M and L´azpita P 2006 Mater. Sci. Eng. A 438–440 445
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twinning and
detwinning in magnesium
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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 015003 (10pp)
doi:10.1088/0953-8984/26/1/015003
Atomic simulations of (101¯ 2), (101¯ 1) twinning and (101¯ 2) detwinning in magnesium Motohiro Yuasa1 , Makoto Hayashi2 , Mamoru Mabuchi2 and Yasumasa Chino1 1
Materials Research Institute for Sustainable Development, National Institute of Advanced Industrial Science and Technology, 2266-98 Anagahora, Shimo-shidami, Moriyama, Nagoya 463-8560, Japan 2 Graduate School of Energy Science, Kyoto University, Yoshidahonmachi, Sakyo, Kyoto 606-8501, Japan E-mail: [email protected] Received 8 August 2013, in final form 8 October 2013 Published 22 November 2013 Abstract
¯ and (1011) ¯ twinning and (1012) ¯ detwinning for Mg were investigated from the (1012) viewpoint of mobility of twinning dislocations and atomic shuffling. First-principles ¯ calculations suggested that the twinning dislocations glide more readily for the (1011) ¯ twinning than for the (1012) twinning. However, this conflicts with the experimental fact of ¯ twin formation. On the other hand, molecular dynamics simulations showed that easier (1012) ¯ twinning than for the (1011) ¯ twinning, the atomic shuffling was more activated for the (1012) which corresponds to the experimental fact. Therefore, it is suggested that the rate-controlling process for the twin formation is the atomic shuffling. Moreover, the calculations and ¯ simulations showed that the twinning dislocations glide more readily for the (1012) ¯ twinning, whereas the atomic shuffling is less activated for the detwinning than for the (1012) detwinning, suggesting that the detwinning occurs easily but is unstable, resulting in easy repetition of twinning–detwinning. (Some figures may appear in colour only in the online journal)
1. Introduction
formed at an initial stage of deformation or in a small strain range, and it increases the uniform elongation [5] and the ¯ twin strain hardening [5–8]. On the other hand, the (1011) occurs at a final stage of deformation or in a large strain range, and it is often related to premature fracture [4, 9]. It is known ¯ that the critical resolved shear stress (CRSS) for the (1012) ¯ ¯ twin is lower than that for the (1011) twin and the (1012) twin ¯ twin [10], although is formed more readily than the (1011) ¯ the boundary energy of the (1012) twin is higher than that ¯ twin [11]. Thus, the degree of difficulty for of the (1011) twin formation in Mg does not depend on the twin-boundary energy. Consequently, one of the problems of deformation twinning in Mg is the rate-controlling process of the twin formation. Recently, the twin nucleation and growth mechanisms in Mg have been investigated by atomic simulations [12–23]. Wang et al [12, 13] showed that a stable nucleus of
Mg alloys are attracting much attention as light structural materials because they exhibit high specific strength and high specific stiffness. However, Mg alloys often exhibit low ductility and low plastic formability as a result of premature fracture [1, 2]. Hence, it is necessary to improve the poor ductility of these alloys in order to increase their applications. Deformation twinning is one of the important deformation modes for plastic deformation in hexagonal close-packed (hcp) metals such as Mg [3], and it strongly affects the mechanical properties. For example, deformation twinning is related to premature fracture [4]. Therefore, it is worthwhile to understand the characteristics of deformation twinning of Mg in order to improve its mechanical properties. The main modes of deformation twins in Mg are ¯ ¯ and (1011)[10 ¯ ¯ twins. The (1012) ¯ twin is (1012)[10 11] 12] 0953-8984/14/015003+10$33.00
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c 2014 IOP Publishing Ltd Printed in the UK
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¯ twin is created by glide of zonal dislocations the (1012) consisting of a partial dislocation and multiple twinning dislocations. The zonal twinning mechanism is energetically favorable relative to the normal twinning mechanism for ¯ the (1012) twinning [13]. Barrett et al [14] suggested that dissociation of matrix dislocations is required for the twin nucleation. Capolungo and Beyerlein [15] proposed a three-dimensional twin nucleation model based on the nonplanar dissociation of a basal partial dislocation by using the continuous linear elastic dislocation theory. Li and ¯ twinning mediated by zonal Ma [16] investigated the (1011) dislocations, which are related to the incomplete dislocation on the twinning plane. Wang et al [17] performed molecular ¯ twinning, and dynamics (MD) simulations of the (1011) they showed that a two-layer twinning dislocation, which is nucleated by the interaction of a basal dislocation with a twin boundary, induces the twin nucleation. These works pointed out the importance of the interactions of twin boundaries with matrix dislocations for the twin nucleation. ¯ However, Capolungo et al [18] noted that the (1012) twin growth is independent of the interaction. Moreover, multiscale modeling studies [19–21] have suggested that the twin nucleation most likely occurs at grain boundaries. Most recently, Xu et al [22] revealed the important role ¯ twinning. Thus, of prismatic/basal interfaces in the (1012) twin formation mechanisms are still in debate. These previous works have investigated the twin nucleation and growth mechanisms through glide of twinning (or zonal) dislocations. However, for hcp metals, the twin formation is not completed just by glide of twinning dislocations but also by atomic movement called atomic shuffling, which is required to complete the twin formation [24]. Zhang et al [25] showed by an experimental investigation with transmission ¯ twin boundaries deviated electron microscopy that the (1012) significantly from the twin plane. This fact suggests that both the atomic shuffling and the glide of twinning dislocations are critical for twinning. Hence, it is important to investigate the twin formation mechanisms in terms of both the glide of twinning dislocations and the atomic shuffling. Another problem related to twinning of Mg is ¯ (1012) detwinning. Many experimental works [26–32] ¯ have demonstrated that the (1012) detwinning occurs readily. For example, Wang et al [27] investigated the effects of compressive predeformation on subsequent tensile deformation of Mg with a ring fiber texture, and they ¯ twinning occurs under compressive found that the (1012) loading and the detwinning occurs during subsequent tensile reloading. Zhang et al [30] showed that twinning–detwinning induced the apparent Bauschinger effect. Hong et al [31] showed that fatigue resistance was improved by tailoring the twinning–detwinning characteristics. Thus, not only the ¯ plane occurs twinning, but also the detwinning of the (1012) readily and it affects the mechanical properties of Mg. If the atomic shuffling is critical for the twinning, it is expected to be critical for the detwinning as well, because the detwinning is not completed just by the glide of twinning dislocations. An aim of the present work is to investigate the mobility of twinning dislocations and the activity of atomic
shuffling in Mg by atomic simulations. Actually, the glide of twinning dislocation and the atomic shuffling may occur simultaneously for the twin formation. In the present work, however, atomic simulations are performed on the assumption that the twinning and the detwinning are complete by the atomic shuffling after the glide of twinning dislocations to investigate the rate-controlling process of the twinning and the detwinning. First-principles calculations of the generalized ¯ stacking fault energy (GSFE) are carried out for the (1012) ¯ twin planes to investigate the twin formation and the (1011) from the viewpoint of the mobility of twinning dislocations. ¯ and (1011) ¯ twinning Then, the atomic shuffling in the (1012) is investigated by relaxing the uncompleted twins after the glide of twinning dislocations using MD simulations, and ¯ twinning is the activity of atomic shuffling for the (1012) ¯ twinning. Moreover, the compared with that for the (1011) mobility of twinning dislocations and the atomic shuffling for ¯ detwinning are compared with those for the (1012) ¯ the (1012) twinning. 2. Calculation methods 2.1. Molecular dynamics simulation
¯ MD simulations were performed on the formation of (1012) ¯ twins through the glide of dislocations and the and (1011) atomic shuffling. Cell models used for the MD simulations are shown in figure 1, where the green circles indicate Mg atoms. Figure 1(a) shows a cell model prior to the glide ¯ twinning, where the x-axis is of dislocations for (1012) ¯ parallel to [1210], the y-axis is parallel to the twinning shear ¯ ¯ direction [1011], and the z-axis is normal to the (1012) plane. Figure 1(b) shows a cell model prior to the glide of ¯ twinning, where the x-axis is parallel dislocations for (1011) ¯ to [1210], the y-axis is parallel to the twinning shear direction ¯ ¯ plane. The [1012], and the z-axis is normal to the (1011) ¯ (1012) unit cell was 1.284 nm long (x-direction), 3.047 nm wide (y-direction), and 6.462 nm high (z-direction), and the ¯ unit cell was 1.284 nm long (x-direction), 2.362 nm (1011) wide (y-direction), and 8.827 nm high (z-direction). Periodic boundaries were adopted along the x- and y-axes. The analytic modified embedded atom method (AMEAM) potential was used. This potential is fit to experimental and ab initio data and reproduces many properties of Mg [33, 34]. The NVT ensemble with a cut-off distance of 0.642 nm was adopted. The integration time step was 1.0 fs throughout the MD simulations. The configurations of atoms were determined by geometry relaxation for 5 ps at 1 K. The temperature was controlled using the Nose–Hoover thermostat technique [35]. The relaxation conditions enabled the stabilization of the ¯ twinning was simulated configuration of atoms. The (1012) based on the zonal twinning mechanism, where the twin 57 ¯ [1011] partial dislocation and one nucleus consisted of one 107 1 ¯ 15 [1011] twinning dislocation [12, 13]. Concretely, the glide of dislocations was imported by the following operations: the atoms of layer 1 and above were displaced along the ¯ [1011] direction over a distance of 0.404 nm, resulting from the glide of a partial dislocation, and then the atoms of 2
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¯ ¯ twinning and (b) (1011) ¯ twinning. The Figure 1. [1210] projection of cell models used for the molecular dynamics simulations, (a) (1012) ¯ ¯ green circles indicate Mg atoms. In (a), the x-axis is parallel to [1210], the y-axis is parallel to the twinning shear direction [1011] and the ¯ plane. In (b), the x-axis is parallel to [1210], ¯ ¯ z-axis is normal to the (1012) the y-axis is parallel to the twinning shear direction [1012] and ¯ plane. the z-axis is normal to the (1011)
¯ layer 3 and above were displaced along the [1011] direction over a distance of 0.049 nm, resulting from the glide of ¯ twinning, first we a twinning dislocation. For the (1011) simulated the importing of twinning dislocation gliding on the ¯ plane based on the zonal twinning mechanism [16]. (1011) However, the twin was not formed through the glide of twinning dislocations and the atomic shuffling. Then, we simulated the glide of twinning dislocations based on the normal twinning mechanism. Namely, the atoms of layer 1 ¯ and above were displaced along the [1012] direction over a distance of 0.260 nm, resulting from the glide of a partial dislocation, and then the atoms of layer 2 and above were ¯ displaced along the [1012] direction over a distance of 0.260 nm, resulting from the glide of a twinning dislocation. ¯ twin As a result of the atomic shuffling after the glide, (1011) formation was completed, as shown later. After importing the dislocations, geometry relaxation was performed at 150–450 K. The atomic shuffling occurred during the relaxation. Figures 2(a) and (b) show the atomic configurations after the glide of dislocations and after the ¯ twinning, and atomic shuffling, respectively, for the (1012) figures 3(a) and (b) show the atomic configurations after the glide of dislocations and after the atomic shuffling,
¯ twinning. Twin formation was respectively, for the (1011) not completed just by the glide of dislocations; the atomic shuffling was also required for the stable structure of the twin. Consequently, twin formation was completed by the atomic ¯ and (1011) ¯ twinning. shuffling for both the (1012) In the present work, the diffusion coefficient was investigated, assuming that the atomic shuffling could be simulated by diffusion. The diffusion coefficient can be calculated by measuring the mean-square displacement (MSD) [36, 37]. According to the Einstein equation for three-dimensional Brownian motion, the diffusion coefficient can be given by D = hλ2 i/6t,
(1)
where D is the diffusion coefficient, hλ2 i is the MSD, and t is the time. The MSD, hλ2 i, can be given by hλ2 i = |1x2 + 1y2 + 1z2 |,
(2)
where 1x is the MSD along the x-axis, 1y is the MSD along the y-axis, and 1z is the MSD along the z-axis. The diffusion 3
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shuffling was investigated in the same manner as the diffusion coefficient. ¯ detwinning was simulated by the reverse The (1012) ¯ twinning. The atoms of layer 3 procedure to the (1012) ¯ direction over a and above were displaced along the [101¯ 1] distance of 0.049 nm (opposite to the twinning dislocation), and then the atoms of layer 1 and above were displaced along ¯ direction over a distance of 0.404 nm (opposite the [101¯ 1] to the partial dislocation). After these procedures, relaxation was performed to induce atomic shuffling at 150–450 K. The diffusion coefficient and activation energy of the atomic ¯ detwinning were calculated in the shuffling for the (1012) ¯ twinning. same manner as for the (1012) 2.2. First-principles calculations
To investigate the mobility of twinning dislocations, firstprinciples calculations were carried out in order to calculate ¯ and (1011) ¯ twin planes. The firstthe GSFE for the (1012) principles calculations were performed using the CASTEP code [38], in which the density functional theory [39, 40] was used with a plane wave basis set to calculate the electronic properties. The exchange–correlation interactions were treated using the generalized gradient approximation within the Perdew–Wang scheme (PW91). The energy cut-off was 350 eV. The ultrasoft pseudopotentials [41] represented in reciprocal space were used in the calculations. The Brillouin zone was sampled using the Monkhorst–Pack 4 × 4 × 1 ¯ and (1011) ¯ planes model, k-point mesh for both the (1012) and the Gaussian smearing with a width of 0.1 eV. The cell models used for the first-principles calculations of GSFE are shown in figure 4. The periodic boundary conditions were adopted along the x-, y-, and z-axes in ¯ both models. Figure 4(a) shows a cell model for the (1012) ¯ twinning, where the x-axis is parallel to [1210], the y-axis ¯ is parallel to the twinning shear direction [1011], and the ¯ ¯ z-axis is normal to the (1012) plane. The (1012) unit cell had periodic lengths of 0.762 nm along the x-axis, 0.642 nm along the y-axis, and 4.60 nm along the z-axis, and the number of the Mg atoms in the cell was 64. A vacuum spacing of 1.5 nm was made along the z-axis. Figure 4(b) is a cell model for ¯ twinning, where the x-axis is parallel to [1210], ¯ the (1011) ¯ the y-axis is parallel to the twinning shear direction [1012], ¯ plane. The (1011) ¯ and the z-axis is normal to the (1011) unit cell had periodic lengths of 1.181 nm along the x-axis, 0.642 nm along the y-axis, and 3.75 nm along the z-axis, and the number of Mg atoms in the cell was 80. A vacuum spacing of 2 nm was made along the z-axis. The GSFE was calculated by displacing one part of the crystal with respect to ¯ ¯ plane and the other along the [1011] direction on the (1012) ¯ ¯ along the [1012] direction on the (1011) plane. The orange and the blue circles in the upper part indicate shifted Mg atoms and the green circles in the lower part indicate non-shifted Mg ¯ twinning, the GSFE was calculated by atoms. For the (1012) ¯ displacing layers 1–9 (orange and blue circles) along [1011] over a distance of 0.404 nm. Then, the GSFE was evaluated ¯ by displacing layers 3–9 (blue circles) along [1011] over a ¯ distance of 0.049 nm. For the (1011) twinning, the GSFE was
¯ twinning, (a) after glide Figure 2. Atomic configurations for (1012) of twinning dislocations and (b) after atomic shuffling.
¯ twinning, (a) after glide Figure 3. Atomic configurations for (1011) of twinning dislocations and (b) after atomic shuffling.
coefficient can be given in another way by QA D = D0 exp − , kB T
(3)
where D0 is the pre-exponential factor, QA is the activation energy, kB is the Boltzmann constant, and T is the absolute temperature. In the present work, the activity of atomic 4
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¯ ¯ twinning and (b) (1011) ¯ twinning. The Figure 4. [1210] projection of cell models used for the first-principles calculations, (a) (1012) orange and the blue circles in the upper part indicate shifted Mg atoms and the green circles in the lower part indicate non-shifted Mg ¯ twinning is calculated by displacing layers 1–9 (orange and blue atoms. The generalized stacking fault energy (GSFE) for the (1012) ¯ ¯ ¯ twinning is evaluated by circles) along [1011], and then by displacing layers 3–9 (blue circles) along [1011]. The GSFE for the (1011) ¯ displacing layers 1–9 (orange and blue circles) along [1012], and then by displacing layers 2–9 (blue circles).
was 3.7 GPa for the glide of a partial dislocation and 5.3 GPa ¯ plane, for the glide of a twinning dislocation for the (1012) and 3.1 GPa for the glide of a partial dislocation and 3.0 GPa ¯ plane. for the glide of a twinning dislocation for the (1011) ¯ It is noted that the maximum slopes for the (1011) plane are ¯ plane. This indicates that the lower than those for the (1012) ¯ plane twinning dislocations glide more easily on the (1011) ¯ than on the (1012) plane. An inspection of figure 5(a) shows that the Burgers vector of the partial dislocation in the present 57 ¯ work is smaller than that by Wang et al [12], that is, 107 [1011], although the reason is unknown. The unstable stacking fault energy in the work by Wang et al is larger than the one in the present work. This supports the idea that the dislocation ¯ plane than on the (1012) ¯ glides more easily on the (1011) ¯ plane. However, it is known experimentally that the (1012) ¯ twin. This cannot twin is formed more easily than the (1011) be explained from the viewpoint of the mobility of twinning dislocations. The movement of atoms from their locations before relaxation to their locations after relaxation at 150 K by the MD simulation is shown in figure 6, where arrows indicate the movement of each atom before and after relaxation. It
evaluated by displacing layers 1–9 (orange and blue circles) ¯ along [1012] over a distance of 0.260 nm. It was then similarly evaluated by displacing layers 2–9 (blue circles). ¯ detwinning, the GSFE was In the case of the (1012) ¯ twinning calculated by the reverse procedure to the (1012) calculations. The GSFE was calculated by displacing layers ¯ direction over a distance of 0.049 nm, 3–9 along the [101¯ 1] ¯ direction over and then displacing layers 1–9 along the [101¯ 1] a distance of 0.404 nm. 3. Results and discussion
¯ ) and (1011 ¯ ) twinning 3.1. (1012 Figure 5 shows the variation in GSFE as a function ¯ and (1011) ¯ planes by the of the displacement in (1012) first-principles calculations. The unstable stacking fault energy and the unstable twinning energy were 25.6 and ¯ plane, and 19.3 ˚ −2 , respectively, for the (1012) 32.5 meV A −2 ¯ plane. Also, ˚ and 32.7 meV A , respectively, for the (1011) the maximum slope [42], characterized by the Peierls stress, 5
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¯ plane and (b) (1011) ¯ plane. The Figure 5. Variation in generalized stacking fault energy as a function of the displacement, (a) (1012) ˚ unstable stacking fault energy and the unstable twinning energy are 25.6 and 32.5 meV A ¯ plane, respectively. ˚ −2 for the (1011) 32.7 meV A
−2
¯ plane, and 19.3 and for the (1012)
Figure 6. Movement of atoms from location of atoms before relaxation to their location after relaxation at 150 K by MD simulation:
¯ plane and (b) (1011) ¯ plane. An arrow indicates the movement of each atom before and after relaxation. (a) (1012)
respectively. The energy of the cell decreased with increasing time in the short-time region less than 2–3 × 10−13 s, reaching a constant at the minimum value in the long-time region. Thus, the atomic movement behavior during relaxation was divided into two regions: the short-time region, where atomic movement was more active, and the long-time region, where atomic movement was less active. In figure 9, the 70% energy reduction point, i.e., the point at which the energy of the cell was reduced by 70%, ¯ twinning and at was reached at 1.0 × 10−13 s for the (1012) −13 ¯ 2.0 × 10 s for the (1011) twinning. The activity of atomic
can be seen that atoms move in various directions during the relaxation, resulting in complete formation of the twin. Similar behavior was obtained under relaxation at 200–450 K. The variations in MSD during relaxation at 150–450 K ¯ and the (1011) ¯ as a function of the time for the (1012) twinning are shown in figures 7 and 8, respectively. The atomic movement tended to be more activated in a short-time region of relaxation, less than approximately 2 × 10−13 s, than in a long-time region. Figures 9(a) and (b) show the variation in energy of the cell as a function of the time ¯ and (1011) ¯ twinning, during relaxation at 1 K for the (1012) 6
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Figure 7. Variations in mean-square displacement during relaxation
Figure 8. Variations in mean-square displacement during relaxation
¯ twinning. at 150–450 K as a function of the time for (1012)
¯ twinning. at 150–450 K as a function of the time for (1011)
¯ twinning, whereas minor differences were seen in the (1011) ¯ and the (1011) ¯ activity of atomic shuffling between the (1012) twinning in the long-time region. In typical atomic diffusion, the diffusion coefficient does not depend on the time. Hence, in the previous works [36, 37], long-time simulations were carried out in order to calculate the diffusion coefficient. In the present work, however, the activity of atomic shuffling depended on the time. This is because the atomic shuffling during relaxation is a result of atomic shuffling required for complete formation of the twin, thermal oscillation of atoms, and migration of the twin boundary. Therefore, it is suggested that the atomic shuffling in the long-time region is related to thermal oscillation of atoms and migration of the twin boundary after complete formation of the twin, and the
shuffling in the short-time region at each temperature was ¯ twinning and at 2.0× calculated at 1.0×10−13 s for the (1012) −13 ¯ 10 s for the (1011) twinning. Because the time needed to reach the 70% energy reduction point was determined at 1 K, the activity of atomic shuffling for complete twin formation at 150–450 K was necessarily in the short-time region. The variations in activity of atomic shuffling in the short- and long-time regions as a function of the inverse temperature for ¯ and (1011) ¯ twinning are shown in figures 10 and the (1012) 11, respectively. In figure 11, the activity of atomic shuffling in the long-time region was calculated at 1.0 × 10−12 s for ¯ twinning and the (1011) ¯ twinning. It is noted both the (1012) that the activity of atomic shuffling in the short-time region ¯ twinning is much larger than that for the for the (1012)
¯ twinning, (b) (1011) ¯ twinning, and Figure 9. Variation in energy of the cell during relaxation at 1 K as a function of the time, (a) (1012) ¯ detwinning. (c) (1012) 7
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Figure 11. Variations in activity of atomic shuffling (D) in the
long-time region as a function of the inverse temperature (1/T) for ¯ and (1011) ¯ twinning. (1012)
Figure 10. Variations in activity of atomic shuffling (D) in the
short-time region as a function of the inverse temperature (1/T) for ¯ and (1011) ¯ twinning. (1012)
¯ ) detwinning 3.2. (1012 atomic shuffling in the short-time region is mainly related to the atomic shuffling required for complete formation of the twin. As shown in figures 7 and 8, the MSD did not reach saturation after complete formation of the twin. This is because migration of the twin boundary occurred after complete formation of the twin. For improved reliability of the simulation, the activity of atomic shuffling was also calculated at times corresponding to the 50% and 90% energy reduction points, and similar trends were obtained. The value of QA for atomic shuffling in the short-time ¯ region was 5.9 meV/atom for the (1012) twinning and ¯ 18.1 meV/atom for the (1011) twinning. O’Handley et al [43] investigated twin-boundary motion in Ni–Mn–Ga and they showed that the activation energy for the twin-boundary motion was 15–30 meV/atom, which was in agreement with the experimental results of the Ni alloy. The activation energies in the short-time region in the present work are comparable to those found by O’Handley et al [43]. The ¯ atomic shuffling was more pronounced for the (1012) ¯ twinning than for the (1011) twinning at all temperatures and the activation energy for atomic shuffling was lower ¯ twinning than for the (1011) ¯ twinning in the for the (1012) ¯ short-time region. Clearly, the atomic shuffling in the (1012) ¯ twinning occurs more easily than in the (1011) twinning. ¯ This corresponds to the experimental fact that the (1012) ¯ twin is formed more easily than the (1011) twin [10]. Thus, the degree of difficulty of twin formation can be explained from the viewpoint of the atomic shuffling. Therefore, it is suggested that the rate-controlling process for complete formation of the twin is the atomic shuffling.
Figure 12 shows the variation in GSFE as a function of the ¯ detwinning. The GSFE increased displacement for the (1012) monotonically as a function of the displacement and the ˚ −2 in displacing layers 3–8 maximum GSFE was 5.5 meV A ¯ where the maximum slope was 3.1 GPa. The along [101¯ 1], GSFE continued to increase until reaching a displacement of 0.23 nm and then it decreased, and consequently the ˚ −2 in displacing layers 1–8 maximum GSFE was 30.7 meV A ¯ It is of interest to note that the maximum slope along [101¯ 1]. ¯ for the detwinning was in displacing layers 3–8 along [101¯ 1] ¯ for the lower than that in displacing layers 3–8 along [101¯ 1] twinning. The atomic shuffling after the glide of dislocations for the detwinning was investigated in the same manner as for the twinning. Because the time at which the 70% energy ¯ reduction point was reached was 1.7 × 10−13 s for (1012) detwinning (figure 9(c)), the activity of atomic shuffling in the ¯ short-time region was calculated at 1.7×10−13 s for the (1012) detwinning. The variation in activity of atomic shuffling as a ¯ detwinning function of the inverse temperature for the (1012) is shown in figure 13. The activation energy for atomic ¯ detwinning shuffling in the short-time region for the (1012) ¯ was 13.4 meV/atom, which is larger than that for the (1012) twinning, and also the activity of atomic shuffling in the short-time region for the detwinning was lower than that for the twinning. Hence, the atomic shuffling was less activated ¯ detwinning than for the (1012) ¯ twinning. for the (1012) The experiments showed that twinning–detwinning behavior occurs repeatedly by cyclic loading [28, 29, 31] and 8
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Figure 12. Variation in generalized stacking fault energy as a
¯ detwinning. function of the displacement for (1012)
that the CRSS for the detwinning is lower than that for the twinning [28, 30]. The present work revealed that twinning dislocations glide more easily for the detwinning, whereas the atomic shuffling is less activated, as compared to the twinning. Therefore, it is suggested that the detwinning occurs readily but it is unstable, resulting in a lower CRSS for the detwinning and the easy repeat of twinning–detwinning.
Figure 13. Variation in activity of atomic shuffling (D) in the
short-time region as a function of the inverse temperature (1/T) for ¯ detwinning. (1012)
4. Conclusions
¯ detwinning than for the (1012) ¯ twinning, whereas (1012) the atomic shuffling was less activated for the detwinning. Therefore, it is suggested that the detwinning occurs easily but it is unstable, resulting in a lower CRSS for the detwinning and easy repeat of twinning–detwinning.
¯ and (1011) ¯ twinning and (1012) ¯ detwinning for Mg (1012) were investigated in terms of the mobility of twinning dislocations and the atomic shuffling. First-principles calculations of the GSFE were carried out in order to investigate the mobility of twinning dislocations. Moreover, MD simulations were performed in order to investigate the atomic shuffling. The results are summarized as follows.
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(1) Twin formation was accomplished not only by the glide of twinning dislocations but also by the atomic shuffling for ¯ and the (1011) ¯ twinning. both the (1012) (2) The GSFE and its maximum slope for the glide of ¯ twinning twinning dislocations were larger for the (1012) ¯ than for the (1011) twinning. This conflicts with the ¯ twin is formed more experimental fact that the (1012) ¯ easily than the (1011) twin. Thus, the degree of difficulty for twin formation could not be explained from the viewpoint of the mobility of twinning dislocations. ¯ (3) The activity of atomic shuffling was larger for the (1012) ¯ twinning and the activation twinning than for the (1011) ¯ energy for atomic shuffling was lower for the (1012) ¯ twinning than for the (1011) twinning. Therefore, it is suggested that the rate-controlling process for twin formation is the atomic shuffling after the glide of twinning dislocations. (4) Also, the calculations and simulations showed that the twinning dislocations tended to glide more readily for the 9
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