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Effects of structural relaxation on the generalized stacking fault energies of hexagonal-close-packed system from first-principles calculations
Computational Materials Science 98 (2015) 405–409
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Effects of structural relaxation on the generalized stacking fault energies of hexagonal-close-packed system from first-principles calculations Yuchen Dou a, Jing Zhang a,b,⇑ a b
College of Materials Science and Engineering, Chongqing University, Chongqing 400044, China National Engineering Research Center for Magnesium Alloys, Chongqing 400044, China
a r t i c l e
i n f o
Article history: Received 3 August 2014 Received in revised form 11 November 2014 Accepted 19 November 2014
Keywords: First-principle calculation Magnesium alloys Defects in solids Stacking fault Relaxation Degrees of freedom
a b s t r a c t In this paper, the effects of relaxation parameters on the first-principle-calculated generalized stacking fault energy (GSFE) were investigated. Two relaxing directions were considered, out-of-plane (N-direction) and in-plane (P-direction). N-direction is normal to the slip plane. P-direction is parallel to the slip plane and perpendicular to the slip direction. The results show that relaxation along the N-direction is essential, especially for the high-index slip plane; relaxation along the P-direction is needed when the atoms on the two sides of the slip direction are unsymmetrical. Discussions were made based on the first-principle calculated forces and the geometry of the atomic configurations of different slip systems. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction The GSFE, introduced by Vítek in 1986 [1], is a fundamental parameter intimately related to the mechanical behavior of a material. It stems from the relative slip between two adjacent atomic planes during the shear deformation of a given slip system. Particularly, the local minimum/maximum of the GSFE is called stable/unstable stacking fault energy (abbreviated as cSF and cUSF in the following). The GSFE can be used to model a large number of atomic-scale phenomena, such as intrinsic ductility based on the Peierls concept [2], solid-solution strengthening and the thermal cross-slip stress of dislocations [3,4]. Successful modeling is highly dependant on the precision of the obtained GSFE values. Since experimental measurement of GSFE is almost impossible, the first-principle method has been widely used in various cases, such as metallic systems (Al [5], Fe [6], Mg [7], Mo [8], Ni [9]) and compounds (GaN [10], Al2O3 [11]). In calculating the GSFE, a procedure called lattice relaxation must be carried out before the calculation of a supercell’s total energy. A survey of literatures reveals that the calculated GSFE is sensitive to the degree of freedom adopted in the relaxation procedure. ⇑ Corresponding author at: College of Materials Science and Engineering, Chongqing University, Chongqing 400044, China. Tel.: +86 23 65111167; fax: +86 23 65102821. E-mail address: [email protected] (J. Zhang). http://dx.doi.org/10.1016/j.commatsci.2014.11.041 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.
Taking magnesium alloys as a prototype, which have attracted significant interest in recent years due to the need for weight reduction in automobile and aerospace industries. Commonly, magnesium alloys have poor room temperature ductility which stems from their hexagonal close packed (HCP) crystal structure. The easiest basal slip system provides only two independent slip systems, far from enough to meet the von Mises requirement for five independent slip systems [12]. Early experiments on pure Mg single crystals have demonstrated that the critical resolved shear stress (CRSS) of the prismatic hai and pyramidal hc + ai slip systems are 50–100 times larger than that of the basal slip system at room temperature; as a consequence, the activation of non-basal slip systems is harder. Fortunately, the CRSS can be modified by alloying elements. It is reported that addition of Zn and Al promotes the activation of the prismatic hai slip [13], and addition of Li eases the activation of the pyramidal hc + ai slip [14]. In this context, it is supposed that the lowered CRSS results from the decreased cUSF caused by alloying element addition. To design new magnesium alloys with satisfactory ductility, the effects of alloying elements on the GSFE has largely been studied by means of first-principle calculation in recent years. Published GSFE data [15–25] (by no means complete) for pure magnesium by first-principle calculation are listed in Table 1, where slip systems {0 0 0 1}h1 1 0 0i (for staking fault I2), {1 1 0 0}h1 1 2 0i and {1 1 2 2}h1 1 2 3i are involved. For the {0 0 0 1}h1 1 0 0i slip system, the published data coincide well with
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each other. The cSF (located at 1.0b; b is the Burgers vector) falls into the range from 33.8 to 48.2 mJ m2; the cUSF (located at 0.5b) falls into the range from 84 to 99 mJ m2. However, for the {1 1 0 0}h1 1 2 0i and {1 1 2 2}h1 1 2 3i slip systems, the published GSFEs show significant discrepancies. In the case of {1 1 0 0}h1 1 2 0i, the cUSF (located at 0.5b) falls into the range from 189 to 354 mJ m2 (Note that there does not exist a cSF in this slip system). For the {1 1 2 2}h1 1 2 3i slip system, there are two cUSF. In this paper, only the larger one of the two cUSF was discussed, whose value shows a much larger discrepancy than the smaller one, as listed in Table 1. The published cSF and cUSF fall into the range from 221 to 399 mJ m2 and from 463 to 1080 mJ m2, respectively. Meanwhile, even the location of the cSF exhibits disagreement. In detail, Wen [17] and Pei [21] proposed that the cSF is located at 0.4b; Ghazisaeidi [18] and Nogaret [22] suggested that the cSF is located at 0.33b; Wang [24] reported that the cSF is located at 0.5b. Experimentally, transmission electron microscopy (TEM) observation illustrate that the second-order pyramidal hc + ai dislocation exists in a dissociated form, i.e. two 1/2 h1 1 2 3i partial dislocations with a {1 1 2 2} stacking fault in between [14], providing strong support that the cSF must be located at 0.5b. Toward this end, Wang [24] got the right result. The striking disparity in both the calculated GSFE value and its location is attributed to the different degree of freedom adopted in the relaxation procedure of first-principle calculation. However, the effects of relaxation parameters on the accuracy of the calculated GSFE have not been recognized and addressed. The {1 1 0 0}h1 1 2 0i and {1 1 2 2}h1 1 2 3i slip systems play very important roles in the plastic deformation process of magnesium alloys. However, published GSEFs exhibit significant discrepancies. To set a benchmark for the GSFE calculations of Mg-based alloy systems in the future, it is necessary and timely to clarify
the effects of relaxation parameters on the accuracy of the GSFE calculation. 2. Computational details First principle calculations were carried out using the Vienna Ab initio simulation package (VASP) [26,27], with the generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE) form [28]. The cut-off energy was set as 300 eV. The first order Methfessle–Paxton with smearing of 0.2 eV was used for structural relaxation until the total energy changes within 106 eV. Then the total energy calculation was performed using linear tetrahedron method with Blöchl correction; the Hellmann–Feynman force was calculated by first order Methfessle–Paxton with smearing of 0.2 eV. For {0 0 0 1}h1 1 0 0i, {1 1 0 0}h1 1 2 0i and {1 1 2 2} h1 1 2 3i slip systems, supercells with 48 atoms were constructed, as illustrated in Fig. 1a, d and g. (In this work, effects of relaxation parameters were discussed under the same supercell size, although different supercell size does affect the calculated GSFE [29].) The Brillouin zone was sampled using a Monkhorst–Pack k-point mesh as following: 7 8 3, 7 4 3 and 4 7 3 for {0 0 0 1} h1 1 0 0i, {1 1 0 0}h1 1 2 0i and {1 1 2 2}h1 1 2 3i, respectively. The slip process was simulated by gradually displacing the upper 6 atom layers with respect to the remaining 6 layers along b (Burgers vector). GSFE was derived with the following equation:
cGSFE ¼ ðEn E0 Þ=A
ð1Þ
where En is the energy of the supercell with a displacement and E0 is the energy of the original supercell, and A is the supercell’s crosssectional area. In this study, two relaxing directions were considered, out-of-plane (N-direction) and in-plane (P-direction). N-direction
Table 1 Calculated first-principle (within the GGA) values of stable stacking fault energy (cSF) and unstable stacking fault energy (cUSF) for pure Mg, in mJ m2. For the {0 0 0 1}h1 1 0 0i slip system, the cSF and cUSF are located at 1.0b and 0.5b, respectively (b is the Burgers vector). For the {1 1 0 0}h1 1 2 0i slip system, there does not exist a stable value and the cUSF is located at 0.5b. For the {1 1 2 2}h1 1 2 3i slip system, there are two cUSF and one cSF, located at different positions depending on the degree of freedom adopted in the relaxation procedures. Values without references are calculated in this work. Relaxation procedure
0
1
{0 0 0 1}h1 1 0 0i
{1 1 0 0}h1 1 2 0i
cUSF
cUSF
cUSF
cSF
cUSF
33.8a 48.2b 34c 36e 36g 34i
87.6a 99c
354c
(0.27b) 452c (0.25b) 240d
(0.42b) 399c (0.33b) 235d
(0.67b) 1080c (0.68b) 475d
92e 94f 84g 92i
189f 212g 218i
(0.3b) 318g (0.27b) 243h
(0.4b) 298g (0.33b) 236h
(0.7b) 559g (0.68b) 485h
351j 356k 288 169 169 169 169
(0.3b) 378j (0.3b) 376k (0.25b) 394 (0.25b) 246 (0.3b) 244 (0.3b) 243 (0.3b) 242
(0.5b) 223j (0.5b) 221k (0.35b) 376 (0.35b) 236 (0.5b) 216 (0.5b) 184 (0.5b) 182
(0.7b) (0.7b) (0.7b) (0.7b) (0.7b) (0.7b) (0.7b)
2 3 4 5 6 7
{1 1 2 2}h1 1 2 3i
cSF
36 35 35 35 35
93 86 86 86 86
466j 463k 1029 503 413 393 390
Note: 0 – Ambiguous relaxation process. 1 – All atoms were fully relaxed along the N-direction. 2 – Atoms in the 6th and 7th planes were relaxed along the P-direction. Relaxation along the N-direction is ambiguous. 3 – All atoms are fixed. 4 – All atoms are fully relaxed along the N-direction. 5 – All atoms are fully relaxed along the Ndirection. Atoms in the 6th and 7th planes are relaxed along the P-direction. 6 – All atoms are fully relaxed along the N-direction. Atoms in the 5th, 6th, 7th and 8th planes are relaxed along the P-direction. 7 – All atoms are fully relaxed along the N-direction and the P-direction. For 1 and 2, the relaxation of supercell’s shape and volume is ambiguous; from 3 to 7, supercell’s shape and size are fully relaxed. a Ref. [15]. b Ref. [16]. c Ref. [17]. d Ref. [18]. e Ref. [19]. f Ref. [20]. g Ref. [21]. h Ref. [22]. i Ref. [23]. j Ref. [24]. k Ref. [25].
Y. Dou, J. Zhang / Computational Materials Science 98 (2015) 405–409
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Fig. 1. (a–c) Atomic configurations for the {0 0 0 1}h1 1 0 1i slip system. (d–f) Atomic configurations for the {1 1 0 0}h1 1 2 0i slip system. (g–i) Atomic configurations for the {1 1 2 2}h1 1 2 3i slip system, Z is perpendicular to both [1 0 1 0] and [1 2 1 3] directions. In all three cases, a vacuum width of 10 Å is added to avoid the interactions due to periodic images. The color strip at the left shows the force (eV/Å) along the N-direction. Comparison between the fixed case and relaxed case shows that relaxation along Ndirection eliminates the force and displaces the stressed atoms along this direction. Since the supercell’s shape and size are allowed to relax, inter-plane spacing expands slightly even when the atoms are fixed. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
is normal to the slip plane. P-direction is parallel to the slip plane and perpendicular to the slip direction. So, when the slip direction is set along the y-direction, P-direction relaxing means relaxing along the x-direction only. 3. Results
lated cSF and cUSF are 36 and 93 mJ m2, respectively. Fully relaxing along the N-direction decreases them to 35 and 86 mJ m2. Fully relaxing along the P-direction does not cause any further influence, as shown in Fig. 2a. Overall, the results reveal that the calculated {0 0 0 1}h1 1 0 0i GSFEs exhibit a weak relaxation parameter dependence, corresponding to the fact that different relaxation parameters result in very similar values.
3.1. {0 0 0 1}h1 1 0 0i slip system 3.2. {1 1 0 0}h1 1 2 0i slip system Calculated SFEs (for staking fault I2) and adopted relaxation parameters are listed in Table 1. As listed in Table 1, for the {0 0 0 1}h1 1 0 0i slip system, when all atoms were fixed, the calcu-
For the {1 1 0 0}h1 1 2 0i slip system, there are two different inter-plane spacing along the P-direction, one is 1.74 Å and the
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Y. Dou, J. Zhang / Computational Materials Science 98 (2015) 405–409
Fig. 2. GSFE curves for (a) {0 0 0 1}h1 1 0 0i, (b) {1 1 0 0}h1 1 2 0i, and (c) {1 1 2 2}h1 1 2 3i slip systems under various relaxation procedures 3, 4, 5, 6 and 7 (see Table 1 for detailed relaxation parameters).
other is 0.84 Å, as shown in Fig. 1d. In this study, relative slip was performed on the larger one. When all atoms were fixed, the calculated cUSF is 288 mJ m2. Fully relaxing along the N-direction decreases it to 169 mJ m2, coinciding with the value calculated by Yuasa (189 mJ m2) [20]. Fully relaxing along the P-direction does not cause any further influence. 3.3. {1 1 2 2}h1 1 2 3i slip system In the case of the {1 1 2 2}h1 1 2 3i slip system, different relaxation parameters influence not only the values of the cSF and cUSF, but also their locations, as shown in Fig. 2c. Therefore, location is also indicated (in parentheses) when referring to the value thereafter. When all atoms are fixed, the calculated cSF and cUSF are (0.35b) 376 and (0.7b) 1029 mJ m2, coinciding with the results in value reported by Wen, i.e. (0.42b) 399 and (0.67b) 1080 mJ m2 [17], though the locations show some discrepancies. Fully relaxing along the N-direction decreases them to (0.35b) 236 and (0.7b) 503 mJ m2, coinciding with the data calculated by Nogaret [22], i.e. (0.33b) 236 and (0.68b) 485 mJ m2 (the slight difference in the locations might stem from different displacing increment in the slip process). It should be noted that relaxation along the Ndirection does not influence the locations of the cSF and cUSF. Allowing two atom planes to relax along the P-direction results in (0.5b) 216 and (0.7b) 413 mJ m2, respectively. Allowing four atom planes to relax along the P-direction decreases them to (0.5b) 184 and (0.7b) 393 mJ m2. Relaxing more atom planes (more than four) along the P-direction might not cause any further changes, since the difference between relaxing four and twelve (fully relaxing along the P-direction) atom planes is negligible, only in the scale of 3 mJ m2. Moreover, the results show that relaxation along the P-direction is the key requirement to find the correct positions of cSF and cUSF. The reason will be discussed later. Finally, it should be mentioned that our previous research predicted the intrinsic ductility of Mg-based binary system successfully. In that study, supercells were fully relaxed along both the N-direction and P-direction in calculating the GSFEs of the {1 1 2 2}h1 1 2 3i system [30]. 4. Discussion 4.1. Effects of the relaxation along the N-direction For different slip systems, relaxing along the N-direction leads to very different reductions on the calculated GSFEs. Taking the cUSF as an example, the reductions for the {0 0 0 1}h1 1 0 0i, {1 1 0 0}h1 1 2 0i and {1 1 2 2}h1 1 2 3i slip systems are 7, 119 and 577 mJ m2, respectively. In the fixed cases the forces (absolute value) exerting on the atoms in the 6th and 7th planes along the N-direction are 0.11, 0.58 and 1.73 eV/Å for the {0 0 0 1}h1 1 0 0i, {1 1 0 0}h1 1 2 0i and {1 1 2 2}h1 1 2 3i slip systems, respectively, as shown in Fig. 1 (visualized with OVITO [31]). These forces are associated with the original inter-plane
spacing between the two relative slip atomic planes, i.e. 2.62, 1.74 and 1.39 Å, for the {0 0 0 1}h1 1 0 0i, {1 1 0 0}h1 1 2 0i and {1 1 2 2}h1 1 2 3i slip systems, respectively. Fully relaxing along the N-direction eliminates the forces and increases these interplane spacing to 2.68, 1.97 and 2.08 Å, leading to lower cUSF values. Based on the discussions, the authors suggest that when atoms are fixed along the N-direction, smaller original inter-plane spacing along this direction results in bigger forces. In other words, relaxation along the N-direction is essential, especially for the highindex slip plane. 4.2. Effects of the relaxation along the P-direction For the {0 0 0 1}h1 1 0 0i and{1 1 0 0}h1 1 2 0i slip systems, relaxation along the P-direction causes no effect. Relaxation procedures 4, 5, 6 and 7 (Table 1) lead to exactly the same GSFE curves, as shown in Fig. 2a and b. However, the {1 1 2 2}h1 1 2 3i slip system is sensitive to the relaxation along the P-direction, which probably stems from the atomic configurations of the two relative slip planes, as shown in Fig. 3. It can be seen that for the {0 0 0 1}h1 1 0 0i and {1 1 0 0}h1 1 2 0i systems, the atoms on the two sides of the slip direction (illustrated with blue arrows) are symmetrical. As a result, the total force along the P-direction should be zero. Whereas for the {1 1 2 2}h1 1 2 3i slip system, the atoms on the two sides of the slip direction are unsymmetrical. During the slip process, a force along the P-direction arises due to the unsymmetrical feature. Taking the site for the cSF as an example, atoms of type ‘A’ should suffer a force along the [1 0 1 0] direction while atoms of type ‘B’ received a force along the opposite direction, when the relaxation along P-direction is inhibited. For the {1 1 2 2}h1 1 2 3i slip system, allowing the atoms to relax along the P-direction locates the position of the cUSF correctly, i.e. at the site of 0.5b. To figure out what happens under this condition, the forces along the [1 0 1 0] direction are illustrated in Fig. 4. For the fixed case, the forces (absolute value) exerting on the atoms in the 6th and 7th planes along the P-direction is 0.5 eV/Å. Allowing two atom planes (6th and 7th) to relax along the P-direction eliminate these forces and displace these atoms along the direction of these forces. Meanwhile, another set of forces with value of 0.25 eV/Å arise (exerting on the atoms in the 5th and 8th planes), which can in turn be eliminated by allowing four atom planes (5th, 6th, 7th and 8th) to relax along the P-direction. When four atom planes are relaxed along the P-direction, the maximum force along this direction decreases to a negligible value, 0.01 eV/Å, coinciding with the fact that allowing four and twelve (fully relaxed) atom planes to relax gets almost the same GSFEs. In conclusion, when the atoms on the two sides of the slip direction are unsymmetrical, relaxation along this direction is needed to eliminate the forces caused by the unsymmetrical atomic configuration. We next turn to the slip path of the {1 1 2 2}h1 1 2 3i slip system. Allowing the atoms to relax along the P-direction can indeed
Y. Dou, J. Zhang / Computational Materials Science 98 (2015) 405–409
Fig. 3. Schematic illustration of the two relative slip atomic planes for (a) the {0 0 0 1}h1 1 0 0i slip system, (b) the {1 1 0 0}h1 1 2 0i slip system, and (c) the {1 1 2 2}h1 1 2 3i slip system. For the {0 0 0 1}h1 1 0 0i and {1 1 0 0}h1 1 2 0i slip systems, the atoms on the two sides of the slip direction (illustrated with blue arrows) are symmetrical, thus the total force along P-direction should be zero. For the {1 1 2 2}h1 1 2 3i slip system, the atoms on the two sides of the slip direction are unsymmetrical, leading to a force along P-direction, marked with red arrows. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Atomic configuration for the {1 1 2 2}h1 1 2 3i slip system with a displacement of 0.5b, projected along the [1 2 1 3] direction. (a) All atoms are fixed along the P-direction. (b) Atoms in the 6th and 7th planes are relaxed along the P-direction. (c) Atoms in the 5th, 6th, 7th and 8th planes are relaxed along the P-direction. (d) All atoms are fully relaxed along the P-direction. The color strip at the left shows the force (eV/Å) along [1 0 1 0] direction. Comparison between the fixed case and relaxed case shows that relaxation along P-direction eliminates the force and displaces the stressed atoms along this direction. The stressed atoms are marked with an ellipse in (a). After the relaxation along the P-direction, these atoms are displaced along this direction, marked with an ellipse in (b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. Atomic configuration for the {1 1 2 2}h1 1 2 3i slip system with a displacement of 0.5b, projected along the [1 2 1 3] direction. (a) All atoms are fixed along the P-direction. (b) Atoms in the 6th and 7th planes are relaxed along the P-direction. (c) Atoms in the 5th, 6th, 7th and 8th planes are relaxed along the P-direction. (d) All atoms are fully relaxed along the P-direction. The color strip at the left shows the displacement (Å) along the [1 0 1 0] direction. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
search out the minimum energy path; on the other hand, is the actual slip direction when following the minimum energy path still along the h1 1 2 3i direction? To clarify this point, we would like firstly to recall a concept ‘‘dislocation core’’. It is well known that atoms in the dislocation core are deviated from the lattice sites;
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beyond the dislocation core, atoms stay on the lattice sites. A convenient way to discuss the slip path of the {1 1 2 2}h1 1 2 3i slip system is to introduce a concept ‘‘stacking fault core’’. It is understandable that atoms in the stacking fault core are deviated from the lattice sites; beyond the stacking fault core, atoms stay on the lattice sites. As illustrated in Fig. 5, allowing relaxation along the P-direction only makes the atoms on the 5th, 6th, 7th and 8th (stacking fault core) atomic planes deviate from the lattice sites. Beyond the stacking fault core, atoms are not influenced by the relaxation along the P-direction, as shown in Fig. 5d, in which all atoms are fully relaxed along the P-direction. Therefore, for the {1 1 2 2}h1 1 2 3i slip system, allowing relaxation along the Pdirection does not change the slip path. The slip path is still along the h1 1 2 3i direction. Relaxation along the P-direction only results in a relaxed stacking fault core, which is a key step to find out the right local minimum of the GSFE. 5. Conclusions The effects of relaxation parameters on the accuracy of the calculated GSFE of Mg were studied in detail. In the fixed case, narrower inter-plane spacing (high-index slip plane) results in larger force along the N-direction, leading to higher GSFE; the unsymmetrical feature of atomic configuration on the two sides of the slip direction causes a force along the P-direction, which influences not only the value but also the location of the calculated GSFE. Thus, relaxation along the N-direction is essential, especially for the high-index slip plane. Relaxation along the P-direction is needed when the atoms on the two sides of the slip direction are unsymmetrical. These findings are also applicable to the calculation of GSFE in other hexagonal close packed (HCP) systems such as Co, Ti, Zn and Zr. Acknowledgement The authors are grateful for the financial support from the National Natural Science Foundation of China (Nos. 51271207 and 51471038). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
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Effects of structural relaxation on the generalized stacking fault energies of hexagonal-close-packed system from first-principles calculations Yuchen Dou a, Jing Zhang a,b,⇑ a b
College of Materials Science and Engineering, Chongqing University, Chongqing 400044, China National Engineering Research Center for Magnesium Alloys, Chongqing 400044, China
a r t i c l e
i n f o
Article history: Received 3 August 2014 Received in revised form 11 November 2014 Accepted 19 November 2014
Keywords: First-principle calculation Magnesium alloys Defects in solids Stacking fault Relaxation Degrees of freedom
a b s t r a c t In this paper, the effects of relaxation parameters on the first-principle-calculated generalized stacking fault energy (GSFE) were investigated. Two relaxing directions were considered, out-of-plane (N-direction) and in-plane (P-direction). N-direction is normal to the slip plane. P-direction is parallel to the slip plane and perpendicular to the slip direction. The results show that relaxation along the N-direction is essential, especially for the high-index slip plane; relaxation along the P-direction is needed when the atoms on the two sides of the slip direction are unsymmetrical. Discussions were made based on the first-principle calculated forces and the geometry of the atomic configurations of different slip systems. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction The GSFE, introduced by Vítek in 1986 [1], is a fundamental parameter intimately related to the mechanical behavior of a material. It stems from the relative slip between two adjacent atomic planes during the shear deformation of a given slip system. Particularly, the local minimum/maximum of the GSFE is called stable/unstable stacking fault energy (abbreviated as cSF and cUSF in the following). The GSFE can be used to model a large number of atomic-scale phenomena, such as intrinsic ductility based on the Peierls concept [2], solid-solution strengthening and the thermal cross-slip stress of dislocations [3,4]. Successful modeling is highly dependant on the precision of the obtained GSFE values. Since experimental measurement of GSFE is almost impossible, the first-principle method has been widely used in various cases, such as metallic systems (Al [5], Fe [6], Mg [7], Mo [8], Ni [9]) and compounds (GaN [10], Al2O3 [11]). In calculating the GSFE, a procedure called lattice relaxation must be carried out before the calculation of a supercell’s total energy. A survey of literatures reveals that the calculated GSFE is sensitive to the degree of freedom adopted in the relaxation procedure. ⇑ Corresponding author at: College of Materials Science and Engineering, Chongqing University, Chongqing 400044, China. Tel.: +86 23 65111167; fax: +86 23 65102821. E-mail address: [email protected] (J. Zhang). http://dx.doi.org/10.1016/j.commatsci.2014.11.041 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.
Taking magnesium alloys as a prototype, which have attracted significant interest in recent years due to the need for weight reduction in automobile and aerospace industries. Commonly, magnesium alloys have poor room temperature ductility which stems from their hexagonal close packed (HCP) crystal structure. The easiest basal slip system provides only two independent slip systems, far from enough to meet the von Mises requirement for five independent slip systems [12]. Early experiments on pure Mg single crystals have demonstrated that the critical resolved shear stress (CRSS) of the prismatic hai and pyramidal hc + ai slip systems are 50–100 times larger than that of the basal slip system at room temperature; as a consequence, the activation of non-basal slip systems is harder. Fortunately, the CRSS can be modified by alloying elements. It is reported that addition of Zn and Al promotes the activation of the prismatic hai slip [13], and addition of Li eases the activation of the pyramidal hc + ai slip [14]. In this context, it is supposed that the lowered CRSS results from the decreased cUSF caused by alloying element addition. To design new magnesium alloys with satisfactory ductility, the effects of alloying elements on the GSFE has largely been studied by means of first-principle calculation in recent years. Published GSFE data [15–25] (by no means complete) for pure magnesium by first-principle calculation are listed in Table 1, where slip systems {0 0 0 1}h1 1 0 0i (for staking fault I2), {1 1 0 0}h1 1 2 0i and {1 1 2 2}h1 1 2 3i are involved. For the {0 0 0 1}h1 1 0 0i slip system, the published data coincide well with
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each other. The cSF (located at 1.0b; b is the Burgers vector) falls into the range from 33.8 to 48.2 mJ m2; the cUSF (located at 0.5b) falls into the range from 84 to 99 mJ m2. However, for the {1 1 0 0}h1 1 2 0i and {1 1 2 2}h1 1 2 3i slip systems, the published GSFEs show significant discrepancies. In the case of {1 1 0 0}h1 1 2 0i, the cUSF (located at 0.5b) falls into the range from 189 to 354 mJ m2 (Note that there does not exist a cSF in this slip system). For the {1 1 2 2}h1 1 2 3i slip system, there are two cUSF. In this paper, only the larger one of the two cUSF was discussed, whose value shows a much larger discrepancy than the smaller one, as listed in Table 1. The published cSF and cUSF fall into the range from 221 to 399 mJ m2 and from 463 to 1080 mJ m2, respectively. Meanwhile, even the location of the cSF exhibits disagreement. In detail, Wen [17] and Pei [21] proposed that the cSF is located at 0.4b; Ghazisaeidi [18] and Nogaret [22] suggested that the cSF is located at 0.33b; Wang [24] reported that the cSF is located at 0.5b. Experimentally, transmission electron microscopy (TEM) observation illustrate that the second-order pyramidal hc + ai dislocation exists in a dissociated form, i.e. two 1/2 h1 1 2 3i partial dislocations with a {1 1 2 2} stacking fault in between [14], providing strong support that the cSF must be located at 0.5b. Toward this end, Wang [24] got the right result. The striking disparity in both the calculated GSFE value and its location is attributed to the different degree of freedom adopted in the relaxation procedure of first-principle calculation. However, the effects of relaxation parameters on the accuracy of the calculated GSFE have not been recognized and addressed. The {1 1 0 0}h1 1 2 0i and {1 1 2 2}h1 1 2 3i slip systems play very important roles in the plastic deformation process of magnesium alloys. However, published GSEFs exhibit significant discrepancies. To set a benchmark for the GSFE calculations of Mg-based alloy systems in the future, it is necessary and timely to clarify
the effects of relaxation parameters on the accuracy of the GSFE calculation. 2. Computational details First principle calculations were carried out using the Vienna Ab initio simulation package (VASP) [26,27], with the generalized gradient approximation (GGA) in the Perdew–Burke–Ernzerhof (PBE) form [28]. The cut-off energy was set as 300 eV. The first order Methfessle–Paxton with smearing of 0.2 eV was used for structural relaxation until the total energy changes within 106 eV. Then the total energy calculation was performed using linear tetrahedron method with Blöchl correction; the Hellmann–Feynman force was calculated by first order Methfessle–Paxton with smearing of 0.2 eV. For {0 0 0 1}h1 1 0 0i, {1 1 0 0}h1 1 2 0i and {1 1 2 2} h1 1 2 3i slip systems, supercells with 48 atoms were constructed, as illustrated in Fig. 1a, d and g. (In this work, effects of relaxation parameters were discussed under the same supercell size, although different supercell size does affect the calculated GSFE [29].) The Brillouin zone was sampled using a Monkhorst–Pack k-point mesh as following: 7 8 3, 7 4 3 and 4 7 3 for {0 0 0 1} h1 1 0 0i, {1 1 0 0}h1 1 2 0i and {1 1 2 2}h1 1 2 3i, respectively. The slip process was simulated by gradually displacing the upper 6 atom layers with respect to the remaining 6 layers along b (Burgers vector). GSFE was derived with the following equation:
cGSFE ¼ ðEn E0 Þ=A
ð1Þ
where En is the energy of the supercell with a displacement and E0 is the energy of the original supercell, and A is the supercell’s crosssectional area. In this study, two relaxing directions were considered, out-of-plane (N-direction) and in-plane (P-direction). N-direction
Table 1 Calculated first-principle (within the GGA) values of stable stacking fault energy (cSF) and unstable stacking fault energy (cUSF) for pure Mg, in mJ m2. For the {0 0 0 1}h1 1 0 0i slip system, the cSF and cUSF are located at 1.0b and 0.5b, respectively (b is the Burgers vector). For the {1 1 0 0}h1 1 2 0i slip system, there does not exist a stable value and the cUSF is located at 0.5b. For the {1 1 2 2}h1 1 2 3i slip system, there are two cUSF and one cSF, located at different positions depending on the degree of freedom adopted in the relaxation procedures. Values without references are calculated in this work. Relaxation procedure
0
1
{0 0 0 1}h1 1 0 0i
{1 1 0 0}h1 1 2 0i
cUSF
cUSF
cUSF
cSF
cUSF
33.8a 48.2b 34c 36e 36g 34i
87.6a 99c
354c
(0.27b) 452c (0.25b) 240d
(0.42b) 399c (0.33b) 235d
(0.67b) 1080c (0.68b) 475d
92e 94f 84g 92i
189f 212g 218i
(0.3b) 318g (0.27b) 243h
(0.4b) 298g (0.33b) 236h
(0.7b) 559g (0.68b) 485h
351j 356k 288 169 169 169 169
(0.3b) 378j (0.3b) 376k (0.25b) 394 (0.25b) 246 (0.3b) 244 (0.3b) 243 (0.3b) 242
(0.5b) 223j (0.5b) 221k (0.35b) 376 (0.35b) 236 (0.5b) 216 (0.5b) 184 (0.5b) 182
(0.7b) (0.7b) (0.7b) (0.7b) (0.7b) (0.7b) (0.7b)
2 3 4 5 6 7
{1 1 2 2}h1 1 2 3i
cSF
36 35 35 35 35
93 86 86 86 86
466j 463k 1029 503 413 393 390
Note: 0 – Ambiguous relaxation process. 1 – All atoms were fully relaxed along the N-direction. 2 – Atoms in the 6th and 7th planes were relaxed along the P-direction. Relaxation along the N-direction is ambiguous. 3 – All atoms are fixed. 4 – All atoms are fully relaxed along the N-direction. 5 – All atoms are fully relaxed along the Ndirection. Atoms in the 6th and 7th planes are relaxed along the P-direction. 6 – All atoms are fully relaxed along the N-direction. Atoms in the 5th, 6th, 7th and 8th planes are relaxed along the P-direction. 7 – All atoms are fully relaxed along the N-direction and the P-direction. For 1 and 2, the relaxation of supercell’s shape and volume is ambiguous; from 3 to 7, supercell’s shape and size are fully relaxed. a Ref. [15]. b Ref. [16]. c Ref. [17]. d Ref. [18]. e Ref. [19]. f Ref. [20]. g Ref. [21]. h Ref. [22]. i Ref. [23]. j Ref. [24]. k Ref. [25].
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Fig. 1. (a–c) Atomic configurations for the {0 0 0 1}h1 1 0 1i slip system. (d–f) Atomic configurations for the {1 1 0 0}h1 1 2 0i slip system. (g–i) Atomic configurations for the {1 1 2 2}h1 1 2 3i slip system, Z is perpendicular to both [1 0 1 0] and [1 2 1 3] directions. In all three cases, a vacuum width of 10 Å is added to avoid the interactions due to periodic images. The color strip at the left shows the force (eV/Å) along the N-direction. Comparison between the fixed case and relaxed case shows that relaxation along Ndirection eliminates the force and displaces the stressed atoms along this direction. Since the supercell’s shape and size are allowed to relax, inter-plane spacing expands slightly even when the atoms are fixed. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
is normal to the slip plane. P-direction is parallel to the slip plane and perpendicular to the slip direction. So, when the slip direction is set along the y-direction, P-direction relaxing means relaxing along the x-direction only. 3. Results
lated cSF and cUSF are 36 and 93 mJ m2, respectively. Fully relaxing along the N-direction decreases them to 35 and 86 mJ m2. Fully relaxing along the P-direction does not cause any further influence, as shown in Fig. 2a. Overall, the results reveal that the calculated {0 0 0 1}h1 1 0 0i GSFEs exhibit a weak relaxation parameter dependence, corresponding to the fact that different relaxation parameters result in very similar values.
3.1. {0 0 0 1}h1 1 0 0i slip system 3.2. {1 1 0 0}h1 1 2 0i slip system Calculated SFEs (for staking fault I2) and adopted relaxation parameters are listed in Table 1. As listed in Table 1, for the {0 0 0 1}h1 1 0 0i slip system, when all atoms were fixed, the calcu-
For the {1 1 0 0}h1 1 2 0i slip system, there are two different inter-plane spacing along the P-direction, one is 1.74 Å and the
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Fig. 2. GSFE curves for (a) {0 0 0 1}h1 1 0 0i, (b) {1 1 0 0}h1 1 2 0i, and (c) {1 1 2 2}h1 1 2 3i slip systems under various relaxation procedures 3, 4, 5, 6 and 7 (see Table 1 for detailed relaxation parameters).
other is 0.84 Å, as shown in Fig. 1d. In this study, relative slip was performed on the larger one. When all atoms were fixed, the calculated cUSF is 288 mJ m2. Fully relaxing along the N-direction decreases it to 169 mJ m2, coinciding with the value calculated by Yuasa (189 mJ m2) [20]. Fully relaxing along the P-direction does not cause any further influence. 3.3. {1 1 2 2}h1 1 2 3i slip system In the case of the {1 1 2 2}h1 1 2 3i slip system, different relaxation parameters influence not only the values of the cSF and cUSF, but also their locations, as shown in Fig. 2c. Therefore, location is also indicated (in parentheses) when referring to the value thereafter. When all atoms are fixed, the calculated cSF and cUSF are (0.35b) 376 and (0.7b) 1029 mJ m2, coinciding with the results in value reported by Wen, i.e. (0.42b) 399 and (0.67b) 1080 mJ m2 [17], though the locations show some discrepancies. Fully relaxing along the N-direction decreases them to (0.35b) 236 and (0.7b) 503 mJ m2, coinciding with the data calculated by Nogaret [22], i.e. (0.33b) 236 and (0.68b) 485 mJ m2 (the slight difference in the locations might stem from different displacing increment in the slip process). It should be noted that relaxation along the Ndirection does not influence the locations of the cSF and cUSF. Allowing two atom planes to relax along the P-direction results in (0.5b) 216 and (0.7b) 413 mJ m2, respectively. Allowing four atom planes to relax along the P-direction decreases them to (0.5b) 184 and (0.7b) 393 mJ m2. Relaxing more atom planes (more than four) along the P-direction might not cause any further changes, since the difference between relaxing four and twelve (fully relaxing along the P-direction) atom planes is negligible, only in the scale of 3 mJ m2. Moreover, the results show that relaxation along the P-direction is the key requirement to find the correct positions of cSF and cUSF. The reason will be discussed later. Finally, it should be mentioned that our previous research predicted the intrinsic ductility of Mg-based binary system successfully. In that study, supercells were fully relaxed along both the N-direction and P-direction in calculating the GSFEs of the {1 1 2 2}h1 1 2 3i system [30]. 4. Discussion 4.1. Effects of the relaxation along the N-direction For different slip systems, relaxing along the N-direction leads to very different reductions on the calculated GSFEs. Taking the cUSF as an example, the reductions for the {0 0 0 1}h1 1 0 0i, {1 1 0 0}h1 1 2 0i and {1 1 2 2}h1 1 2 3i slip systems are 7, 119 and 577 mJ m2, respectively. In the fixed cases the forces (absolute value) exerting on the atoms in the 6th and 7th planes along the N-direction are 0.11, 0.58 and 1.73 eV/Å for the {0 0 0 1}h1 1 0 0i, {1 1 0 0}h1 1 2 0i and {1 1 2 2}h1 1 2 3i slip systems, respectively, as shown in Fig. 1 (visualized with OVITO [31]). These forces are associated with the original inter-plane
spacing between the two relative slip atomic planes, i.e. 2.62, 1.74 and 1.39 Å, for the {0 0 0 1}h1 1 0 0i, {1 1 0 0}h1 1 2 0i and {1 1 2 2}h1 1 2 3i slip systems, respectively. Fully relaxing along the N-direction eliminates the forces and increases these interplane spacing to 2.68, 1.97 and 2.08 Å, leading to lower cUSF values. Based on the discussions, the authors suggest that when atoms are fixed along the N-direction, smaller original inter-plane spacing along this direction results in bigger forces. In other words, relaxation along the N-direction is essential, especially for the highindex slip plane. 4.2. Effects of the relaxation along the P-direction For the {0 0 0 1}h1 1 0 0i and{1 1 0 0}h1 1 2 0i slip systems, relaxation along the P-direction causes no effect. Relaxation procedures 4, 5, 6 and 7 (Table 1) lead to exactly the same GSFE curves, as shown in Fig. 2a and b. However, the {1 1 2 2}h1 1 2 3i slip system is sensitive to the relaxation along the P-direction, which probably stems from the atomic configurations of the two relative slip planes, as shown in Fig. 3. It can be seen that for the {0 0 0 1}h1 1 0 0i and {1 1 0 0}h1 1 2 0i systems, the atoms on the two sides of the slip direction (illustrated with blue arrows) are symmetrical. As a result, the total force along the P-direction should be zero. Whereas for the {1 1 2 2}h1 1 2 3i slip system, the atoms on the two sides of the slip direction are unsymmetrical. During the slip process, a force along the P-direction arises due to the unsymmetrical feature. Taking the site for the cSF as an example, atoms of type ‘A’ should suffer a force along the [1 0 1 0] direction while atoms of type ‘B’ received a force along the opposite direction, when the relaxation along P-direction is inhibited. For the {1 1 2 2}h1 1 2 3i slip system, allowing the atoms to relax along the P-direction locates the position of the cUSF correctly, i.e. at the site of 0.5b. To figure out what happens under this condition, the forces along the [1 0 1 0] direction are illustrated in Fig. 4. For the fixed case, the forces (absolute value) exerting on the atoms in the 6th and 7th planes along the P-direction is 0.5 eV/Å. Allowing two atom planes (6th and 7th) to relax along the P-direction eliminate these forces and displace these atoms along the direction of these forces. Meanwhile, another set of forces with value of 0.25 eV/Å arise (exerting on the atoms in the 5th and 8th planes), which can in turn be eliminated by allowing four atom planes (5th, 6th, 7th and 8th) to relax along the P-direction. When four atom planes are relaxed along the P-direction, the maximum force along this direction decreases to a negligible value, 0.01 eV/Å, coinciding with the fact that allowing four and twelve (fully relaxed) atom planes to relax gets almost the same GSFEs. In conclusion, when the atoms on the two sides of the slip direction are unsymmetrical, relaxation along this direction is needed to eliminate the forces caused by the unsymmetrical atomic configuration. We next turn to the slip path of the {1 1 2 2}h1 1 2 3i slip system. Allowing the atoms to relax along the P-direction can indeed
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Fig. 3. Schematic illustration of the two relative slip atomic planes for (a) the {0 0 0 1}h1 1 0 0i slip system, (b) the {1 1 0 0}h1 1 2 0i slip system, and (c) the {1 1 2 2}h1 1 2 3i slip system. For the {0 0 0 1}h1 1 0 0i and {1 1 0 0}h1 1 2 0i slip systems, the atoms on the two sides of the slip direction (illustrated with blue arrows) are symmetrical, thus the total force along P-direction should be zero. For the {1 1 2 2}h1 1 2 3i slip system, the atoms on the two sides of the slip direction are unsymmetrical, leading to a force along P-direction, marked with red arrows. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Atomic configuration for the {1 1 2 2}h1 1 2 3i slip system with a displacement of 0.5b, projected along the [1 2 1 3] direction. (a) All atoms are fixed along the P-direction. (b) Atoms in the 6th and 7th planes are relaxed along the P-direction. (c) Atoms in the 5th, 6th, 7th and 8th planes are relaxed along the P-direction. (d) All atoms are fully relaxed along the P-direction. The color strip at the left shows the force (eV/Å) along [1 0 1 0] direction. Comparison between the fixed case and relaxed case shows that relaxation along P-direction eliminates the force and displaces the stressed atoms along this direction. The stressed atoms are marked with an ellipse in (a). After the relaxation along the P-direction, these atoms are displaced along this direction, marked with an ellipse in (b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. Atomic configuration for the {1 1 2 2}h1 1 2 3i slip system with a displacement of 0.5b, projected along the [1 2 1 3] direction. (a) All atoms are fixed along the P-direction. (b) Atoms in the 6th and 7th planes are relaxed along the P-direction. (c) Atoms in the 5th, 6th, 7th and 8th planes are relaxed along the P-direction. (d) All atoms are fully relaxed along the P-direction. The color strip at the left shows the displacement (Å) along the [1 0 1 0] direction. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
search out the minimum energy path; on the other hand, is the actual slip direction when following the minimum energy path still along the h1 1 2 3i direction? To clarify this point, we would like firstly to recall a concept ‘‘dislocation core’’. It is well known that atoms in the dislocation core are deviated from the lattice sites;
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beyond the dislocation core, atoms stay on the lattice sites. A convenient way to discuss the slip path of the {1 1 2 2}h1 1 2 3i slip system is to introduce a concept ‘‘stacking fault core’’. It is understandable that atoms in the stacking fault core are deviated from the lattice sites; beyond the stacking fault core, atoms stay on the lattice sites. As illustrated in Fig. 5, allowing relaxation along the P-direction only makes the atoms on the 5th, 6th, 7th and 8th (stacking fault core) atomic planes deviate from the lattice sites. Beyond the stacking fault core, atoms are not influenced by the relaxation along the P-direction, as shown in Fig. 5d, in which all atoms are fully relaxed along the P-direction. Therefore, for the {1 1 2 2}h1 1 2 3i slip system, allowing relaxation along the Pdirection does not change the slip path. The slip path is still along the h1 1 2 3i direction. Relaxation along the P-direction only results in a relaxed stacking fault core, which is a key step to find out the right local minimum of the GSFE. 5. Conclusions The effects of relaxation parameters on the accuracy of the calculated GSFE of Mg were studied in detail. In the fixed case, narrower inter-plane spacing (high-index slip plane) results in larger force along the N-direction, leading to higher GSFE; the unsymmetrical feature of atomic configuration on the two sides of the slip direction causes a force along the P-direction, which influences not only the value but also the location of the calculated GSFE. Thus, relaxation along the N-direction is essential, especially for the high-index slip plane. Relaxation along the P-direction is needed when the atoms on the two sides of the slip direction are unsymmetrical. These findings are also applicable to the calculation of GSFE in other hexagonal close packed (HCP) systems such as Co, Ti, Zn and Zr. Acknowledgement The authors are grateful for the financial support from the National Natural Science Foundation of China (Nos. 51271207 and 51471038). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31]
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