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Strain-induced transformation between vacancy voids and stacking fault tetrahedra in Cu
Computational Materials Science 158 (2019) 359–368
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Strain-induced transformation between vacancy voids and stacking fault tetrahedra in Cu
T
Hongbo Xva, Jie Zhaoa, , Fei Yea,b, , Ke Tonga ⁎
a b
⁎
School of Materials Science and Engineering, Dalian University of Technology, 2 Linggong Road, Dalian 116024, Liaoning, China Department of Materials Science and Engineering, Southern University of Science and Technology, No. 1088, Xueyuan Rd, Shenzhen 518055, Guangdong, China
ARTICLE INFO
ABSTRACT
Keywords: Molecular dynamics Atomistic simulation Vacancy cluster Stacking fault tetrahedra Volumetric strain
The transformation between voids and stacking fault tetrahedra (SFTs) under volumetric strain has been systematically investigated by atomistic computer simulation. It has been found out that equiaxial voids or SFTs have high stability under tensile or compressive strain. Molecular dynamic simulations show that the SFTs can transform to voids even at extremely low temperature under constant tensile strain. On the contrary, the equiaxial voids can transform to SFTs under constant compressive strain. The voids and SFTs can also transform to each other under cycling strain. During the transformation from SFTs to voids, four atoms at the vertex of SFT move outward first. Then, the {1 1 1} plane atoms move by layers toward one of the vertices. Finally, the opposing {1 1 1} plane atoms move toward the opposite direction, thereby forming planar voids. The planar voids transform to SFT through the opposite process. This strain-induced transformation is remarkably affected by temperature and strain rate. High temperature can advance the strain of the transformation, while the strain rate exerts opposite effects.
1. Introduction Various defects, including vacancies, self-interstitials, and intersecting stacking faults, are created in materials during irradiation or high-speed deformation [1–3]. Vacancies are predominant point defects, whereas the others anneal out again during the subsequent atomic rearrangements [4–6]. When a supersaturation of vacancies is induced, the vacancies profoundly influence the material properties [2,7–11]. Intensive studies have suggested that vacancies can aggregate to form voids, dislocation loops, and stacking faults [12–16]. Considerable research has been conducted to elucidate the structure of the vacancy cluster. Voids and stacking fault tetrahedra (SFTs) are dominant vacancy clusters in face-centered cubic (fcc) metals [1–3,17]. Voids can transform into SFTs by collapsing into the Frank loops (a disk of vacancies into the {1 1 1} plane), and the Frank loops then dissociate into SFTs [18,19]. Computer simulations have shown that voids collapse into the Frank loops at 800 K [20], and SFTs form around a void at 9000 K [21]. Other mechanisms have also been proposed [22]. Uberuaga showed that voids can transform directly to SFTs without passing through a Frank loop [23]. This transformation proceeds stepwise through a series of 3D structures even at a temperature as low as 400 K. Apart from temperature, stress and strain also play crucial roles in
the formation and growth of vacancy clusters [24–28]. Complicated behavior of the binding energy with uniaxial tensile strain is observed under uniaxial strain [24], and the planar cluster and void become the dominant types under high strain [29]. Under volumetric strain, the formation energy of monovacancies and divacancies monotonously decreases with the decrease in strain in Al [30]. Peng et al. [24] found via molecular static simulation that the formation energy of vacancy clusters with up to 19 vacancies in Cu increases first and then decreases under volumetric tensile strain, whereas the binding energy monotonously increases with the increase in strain, thereby increasing the cluster stability. Moreover, the stability of the clusters weakens as temperature decreases. Such effects of strain and temperature on the stability of vacancy were also showed in the molecular static simulation of Ni [31]. However, most studies on volumetric strain mainly focus on vacancy voids. The effects on the stability of the clusters with other structures, especially SFTs, have not been investigated. Moreover, since the transformation from voids to SFTs consists of a volume and a shape change in vacancy clusters [23], strain may influence the transformation, and inverse transformation is possible under certain strain and temperature condition. Actually, in the previous work of Uberuaga [23], compressive stress may occur as temperature increases under constant volume during simulation.
⁎ Corresponding authors at: School of Materials Science and Engineering, Dalian University of Technology, 2 Linggong Road, Dalian 116024, Liaoning, China (J. Zhao). E-mail addresses: [email protected] (J. Zhao), [email protected] (F. Ye).
https://doi.org/10.1016/j.commatsci.2018.11.026 Received 2 August 2018; Received in revised form 11 November 2018; Accepted 12 November 2018 0927-0256/ © 2018 Elsevier B.V. All rights reserved.
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In the present study, the effects of volumetric strain on the stability of vacancy voids and SFTs in Cu and the transformation between them are systematically studied by atomistic computer simulation. The relative stability of voids and SFTs under volumetric strain is examined by molecular static simulation. Then, the transformation between voids and SFTs is studied under constant and continuously varied strains by molecular dynamic simulation. The mechanism of the transformation under strain is also discussed.
total energy of the supercell with (N–n) atoms and an n-vacancy cluster, and ENtot is the total energy of the corresponding defect-free supercell containing N atoms under the strain. To compare the stability of the clusters, the binding energy per vacancy for an n-vacancy cluster under a strain was given by b Env =
where
F Env
N n
n N
ENtot
is the formation energy for an n-vacancy cluster,
Envf
(2)
where is the formation energy of an isolated vacancy and is the sum of E1fv . A positive binding energy indicates a preference for the n isolated vacancies to form an n-vacancy cluster, and higher binding energy denotes greater stability. The molecular dynamic calculation was performed under constant and continuously varied strains. The calculations under constant strain were performed at zero, −5%, 4%, and 5% strains. The strain higher than 4% (i.e., 5%) may result in unstable structure of the 15 V and 21 V clusters. The calculations were conducted at 10 K with additional simulation for 6 V at 1 K under zero strain. Simulation was conducted for 6 ps and sampled every 0.01 ps. The calculations under varied strains were performed from −5% to 5% for 6 V and from −5% to 4% for 15 V and 21 V. The temperatures were 300, 400, 500, and 600 K. 300 K was about room temperature, and 600 K was about the recrystallization temperature for Cu. Three strain rates, 1 × 107 s−1, 1 × 108 s−1, and 1 × 109 s−1, were applied.
The atomistic simulation based on empirical interatomic potentials was used in this work. The atomic interaction of Cu was described by an embedded atom model developed by Minish et al. [32]. This model accurately reproduces the atomic structure and energetics for vacancies in Cu [33–37]. The vacancy clusters in bulk fcc Cu were studied using classic molecular static and dynamic approaches implemented in the LAMMPS codes [38]. A cubic supercell containing 4000 (10 × 10 × 10) atoms with periodic boundary conditions along three directions was produced. The supercell was oriented in the cubic directions, that is, x in [1 0 0], y in [0 1 0], and z in [0 0 1]. The variation in the total energy of the system containing the largest cluster as a function of the supercell size was checked to ensure the size was sufficiently large. The vacancy clusters were constructed by removing atoms from the supercells. Specifically, the structure for perfect SFTs, which correspond to atoms surrounded by four {1 1 1} planes forming a regular tetrahedron, was produced by removing an equilateral triangle of atoms from one {1 1 1} plane and forming a planar vacancy cluster. Other atoms of tetrahedron above the planar cluster were collapsed to it along the 〈1 1 1〉/4 direction. Thus, the vacancy numbers of the SFTs were counted from the planar vacancy clusters. Corresponding to the formation of prefect SFTs, the vacancy clusters containing 6, 15, and 21 vacancies (denoted as 6 V, 15 V, and 21 V, respectively) were studied in this work. The most stable configurations of vacancy voids with these vacancy numbers as demonstrated in previous studies [24] are shown in Fig. 1. The 6 V void is octahedron clusters. The larger voids with 15 V and 21 V are considered as combinations of tetrahedral and octahedral clusters [29], and they appear as sphere-like shape [24]. The volumetric strain was made isotropically by applying an equal strain along the x, y, and z directions. In the molecular static calculation, the strain was ranged from −8% (compressive strain) to 8% (tensile strain) with intervals of 0.2%. To describe the energy cost for the formation of a vacancy cluster with respect to the perfect lattice under a strain, the formation energy per vacancy for an n-vacancy cluster was defined as F Env 1 tot = EnvN n n
F Env ) = E1fv
E1fv
2. Methodology
Envf =
1 ( E1fv n
E1fv
3. Results 3.1. Stability of vacancy voids and SFTs under volumetric strain The formation energy of monovacancy under volumetric strain is the basis for the calculation of binding energy. The monovacancy formation energy under zero strain is 1.3091 eV, which agrees very well with previous calculations [24]. Fig. 2 shows that the formation energy increases monotonically with the increase in volumetric strain, which agrees well with previous calculations [29,30] and confirms the validity of the calculation. The stability of the vacancy voids and SFTs was studied to compare their relative stability under strain. Figs. 3 and 4 show the average formation and binding energies under volumetric strain, respectively. As shown in Fig. 3, the formation energies increase monotonically as functions of strain for voids and SFTs consisting of 6–21 V. The increase in the formation energy for the SFTs is faster than that for voids, indicating that the strain exerts more effect on SFT than on void. The voids show lower formation energy under tensile strain, whereas the SFTs show lower formation energy under compressive strain. Thus, the voids or SFTs have higher formation tendency under tensile or compressive strain. The difference in the formation energy at certain
(1) tot Env N n
is the
Fig. 1. Structure of voids with (a) 6 V, (b) 15 V, and (c) 21 V.
Fig. 2. Formation energy for monovacancy as functions of volumetric strain. 360
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Fig. 3. Formation energy per vacancy for voids and SFTs with (a) 6 V, (b) 15 V, and (c) 21 V as functions of volumetric strain.
Fig. 4. Binding energy per vacancy for voids and SFTs with (a) 6 V, (b) 15 V, and (c) 21 V as functions of volumetric strain.
compressive or tensile strain is larger for larger clusters. Moreover, the strain at energy equivalent points in the figure, which are the intersections of the curves of voids and SFTs, are not at zero strain but under compressive strain. As shown in Table 1, the strain at energy equivalent point shifts to compressive strain for larger clusters, indicating that the voids have high formation tendency even under zero strain. In particular, the strain range of the void formation becomes larger as the vacancy number increases. The results of the binding energies are shown in Fig. 4. The binding energies of SFTs decrease with the increase in strain. On the contrary, those for voids increase with the increase in strain. Moreover, the binding energies decrease rapidly when the strain is larger than about
Table 1 Strain at energy equivalent point for 6 V, 15 V, and 21 V voids and SFTs. Vacancy number
6
15
21
Strain (%)
−3.3
−3.9
−4.1
5%. Correspondingly, the supercell structure gradually becomes disordered when the strain is further increased. As mentioned above, the structure of the 15 V and 21 V clusters becomes unstable when the strain is higher than about 4%. The strains at which the binding energies of voids and SFTs are equal are also not at zero strain, which are numerically the same as those of formation energies. 361
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The voids show lower formation energy and higher binding energy under tensile strain, whereas the SFTs show lower formation energy and higher binding energy under compressive strain. Therefore, the voids or SFTs have high stability under tensile or compressive strain. 3.2. Transformation between voids and SFTs under constant strain Since the voids or SFTs have higher formation tendency and stability under tensile or compressive strain, the voids and SFTs may transform between each other under certain strain condition. Therefore, molecular dynamic simulation was then performed under zero, −5%, 4%, and 5% strains, respectively. The results show that the transformation takes place very fast after starting the simulation even under very low temperature. For instance, under zero strain and 1 K, the transformation of 6 V SFT to void only takes 0.3 ps. To compare the effect of strain on structure transformation, the following calculations were conducted at a relatively low temperature of 10 K. The results in Fig. 5 show that most vacancy clusters can transform to a relatively stable structure as expected. Specifically, under −5% strain, the voids transform into SFTs. By contrast, under zero, 4%, and 5% strains, the SFTs transform into voids. The accomplishment of the transformation was determined from the geometries of the cluster structure observed in the visualization software OVITO [39]. The details of the transformation from SFT to void are affected by the cluster size and strain, as shown in Fig. 5. The transformations from the 6 V SFT to void can take place under zero, 4%, and 5% strains (see supplementary video S1). While the 15 V and 21 V SFTs cannot transform to voids under zero strain. When strain increases to 4%, the 15 V and 21 V SFTs transform to planar voids on the {1 1 1} plane (see supplementary videos S2 and S3). As strain increases to 5%, the voids transformed from the 15 V and 21 V SFTs expand gradually, thereby leading to the disorderliness of the supercell structure. Furthermore, the transformation time shows dependence on cluster size. The 6 V SFTs at zero, 4%, and 5% strains take around 0.4 ps to finish transforming into voids. The transformations from the 15 V and 21 V SFTs to voids at 4% strain take less than 4 ps.
supplementary video S2. Transformation from 15 V SFT to void at 10 K under constant strain 4%.
supplementary video S3. Transformation from 21 V SFT to void at 10 K under constant strain 4%.
The atomic scale processes of the transformation from SFT to void are shown in Fig. 6. In the case of 15 V and 21 V clusters (Fig. 6(c)–(f)), the figure shows that four atoms at the vertex of SFT move outward along 〈1 1 1〉 direction first. Then, the atoms on {1 1 1} planes covered by pink polyhedra move by layers toward one of the vertices along [11¯1] direction. Finally, the atoms on the opposing {1 1 1} plane marked by blue surface move along the opposite [1¯11¯] direction. At the end of the migration, a planar void on the {1 1 1} plane is formed (see supplementary videos S4 and S5). The transformation of SFT with 6 V is completed by expanding the four atoms outward along 〈1 1 1〉 direction because only four atoms are present in the center of this SFT (Fig. 6(a) and (b)). supplementary video S1. Transformation from 6 V SFT to void at 10 K under constant strain 4%.
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this range, the structure of the vacancy cluster can compete between void and SFT. Specifically, no such structure competition is observed for 15 V. Fig. 7 also shows that the strain of the start transformation surface increases with the increase in temperature. Thus, the transformation can take place under large strain because the temperature can promote the mobility of the atoms. The strain of the start transformation surface also increases with the decrease in strain rate because low strain rate provides sufficient time for atoms to migrate. However, the strain of finish transformation surface does not show such correlation with temperature and strain rate. The structures before and after the transformation are shown in Fig. 8. The figure shows that equiaxial voids with 6 V, 15 V, and 21 V can transform to perfect SFTs. The initial structure of the voids can maintain its configuration before transformation even under high temperature (see supplementary videos S6 and S7).
supplementary video S4. Atomic details of transformation from 15 V SFT to void at 10K under constant strain 4%.
supplementary video S6. Transformation from 15 V void to SFT at 400 K with 1 × 108 s−1 strain rate from tensile to compressive strain.
supplementary video S5. Atomic details of transformation from 21 V SFT to void at 10K under constant strain 4%.
In previous results of voids transforming into SFTs [23], the transformation occurs at relatively high temperature (higher than 400 K) at constant volume condition. The transformation in the present work takes place at much lower temperature. This difference is possibly due to the large strain in our work. The initial structure of void is equiaxial in the transformation from void to SFT. However, the structure of void transformed from SFT becomes planar in the current work. If the planar void transforms to SFT under compressive strain, then its specific process should be the opposite, as that shown in Fig. 6. This will be further verified by the results of cyclic strain. 3.3. Transformation between voids and SFTs under varied strains The transformation between void and SFT can also takes place under varied strains and is affected by temperature and strain rate. Fig. 7 shows the strains of structure transformation from tensile to compressive strain under different temperatures and strain rates connected into surfaces. In this transformation process, the upper and nether surfaces are start and finish surface for the structure transformation, respectively. As shown in the figure, a strain range between the two surfaces exists in the structure transformation for 6 V and 21 V. In
supplementary video S7. Transformation from 21 V void to SFT at 300 K with 1 × 108 s−1 strain rate from tensile to compressive strain.
Fig. 9 shows the transformation surfaces from compressive to tensile strain under different temperatures and strain rates. In contrast to Fig. 7, the nether surface is the start transformation surface. As shown in the figure, the structures for 15 V and 21 V clusters keep stable after transformation, whereas that for 6 V cluster still shows structure
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Fig. 5. Configurations of vacancy clusters before and after the transformation between voids and SFTs under (a) −5%, (b) 0%, (c) 4%, and (d) 5% constant strains. Atoms are colored on the basis of centrosymmetry parameter (CSP) [9,40]. Atoms with CSP values smaller than 3 are removed for a better view. The same method is also used in Fig. 6(a), (c), and (e) and Figs. 8, 10, and 12. See supplementary video S1, 2 and 3 for details. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
competition in a certain strain range. In addition, the strain of the start transformation surface increases with the decrease in temperature and increases with the increase in strain rate. In this transformation process, the effects of temperature and strain rate are the same as in the transformation from tensile to compressive strain. The structures before and after the transformation of 6 V, 15 V, and 21 V clusters are shown in Fig. 10. The figure shows that the 6 V SFT can transform to equiaxial void, while the 15 V and 21 V SFTs can transform to planar voids on the {1 1 1} plane. Since void and SFT can transform to each other under varied strains, they may transform cyclically under cyclic strain condition. This cyclical transformation is observed when the strain is from −5% to 4% and immediately back to −5% under different temperatures and strain rates (see supplementary videos S8 and S9). The transformations from −5% to 4% are the same as those shown in Figs. 9 and 10. The results of the transformations from 4% to −5% are shown in Figs. 11 and 12. Notably, the initial structure of void is planar in the transformation from void to SFT. As shown in Fig. 11, the transformation surfaces for 15 V and 21 V clusters show no correlation with temperature and strain rate, which is because there are differences between the structures of planar voids transformed from SFTs in the first step.
supplementary video S9. Transformation from 21 V SFT to void and then back to SFT at 400 K with 1 × 108 s−1 strain rate under cycle strain.
4. Conclusions The transformation between voids and SFTs under volumetric strain are systematically investigated by computer simulation. The following conclusions have been obtained: (1) Molecular static calculation shows that equiaxial voids or SFTs have high stability under tensile or compressive strain. Molecular dynamic simulations show that SFTs can transform to voids even at an extremely low temperature of 1 K under constant tensile strain. On the contrary, equiaxial voids can transform to SFTs under constant compressive strain. The transformations take place very fast. For instance, the transformation from 15 V SFT to void at 4% under 10 K only takes 4 ps after starting the simulation. (2) The atomic mechanism of transformation from SFT to void is observed. First, four atoms at the vertex of SFT move outward. Then, the {1 1 1} plane atoms move by layers toward one of the vertices. Finally, the opposing {1 1 1} plane atoms move toward the opposite direction. If the planar void transforms to SFT under compressive strain, then its specific process is opposite.
supplementary video S8. Transformation from 15 V SFT to void and then back to SFT at 400 K with 1 × 108 s−1 strain rate under cycle strain. 364
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Fig. 6. The detailed structures of the clusters for the transformation from (a) 6 V, (c) 15 V, and (e) 21 V SFT to void under 10 K, and their corresponding abstract structures (b), (d), and (f), respectively. The red and blue spheres in the model structure represent the atoms at lattice point and tetrahedral interstices, respectively. The surface in (d) and (f) marks the atoms which will migrate in 〈1 1 1〉 direction. See supplementary videos S4 and S5 for details.
(3) Voids and SFTs can transform to each other under varied strains and can also transform cyclically under cyclic strains. In these processes, temperature and strain rate play important roles. High temperature can ease structure transformation by increasing the mobility of atoms. Meanwhile, low strain rate can provide plenty of
time for the migration of atoms. Consequently, higher temperature and lower strain rate can promote the transformation and advance the strain of transformation.
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Fig. 7. Transformation surfaces at which the equiaxial voids with (a) 6 V, (b) 15 V, and (c) 21 V transform to SFTs from tensile to compressive strain.
Fig. 10. Structures of vacancy cluster before and after the transformation from SFTs with (a) 6 V, (b) 15 V, and (c) 21 V to voids when the strain varied from compression to tension. The voids transformed from SFTs with 15 V and 21 V are on the {1 1 1} planes.
Fig. 8. Structures of vacancy cluster before and after the transformation from equiaxial voids with (a) 6 V, (b) 15 V, and (c) 21 V to SFTs when the strain varied from tension to compression. See supplementary videos S6, S7 for details.
Fig. 9. Transformation surfaces at which the SFTs with (a) 6 V, (b) 15 V, and (c) 21 V transform to {1 1 1} plane voids from compressive to tensile strain.
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Fig. 12. Structures of vacancy cluster before and after the transformation from voids with (a) 6 V, (b) 15 V, and (c) 21 V to SFTs under cycle strain. The structure of voids is the same as that shown in Fig. 10. See supplementary videos S8, S9 for details.
5. Data availability The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations. CRediT authorship contribution statement Hongbo Xv: Investigation, Formal analysis, Writing - original draft. Jie Zhao: Funding acquisition. Fei Ye: Writing - review & editing. Ke Tong: Methodology, Conceptualization. Acknowledgement This work was supported by the National Natural Science Foundation of China [grant numbers 51771042, U1610256] References [1] M. Kiritani, Story of stacking fault tetrahedra, Mater. Chem. Phys. 50 (1997) 133–138. [2] S.J. Zinkle, L.E. Seitzman, W.G. Wolfer, I. Energy calculations for pure metals, Philos. Mag. A 55 (1987) 111–125. [3] S.J. Zinkle, W.G. Wolfer, G.L. Kulcinski, L.E. Seitzman, II. Effect of oxygen and helium on void formation in metals, Philos. Mag. A 55 (1987) 127–140. [4] M.R. Sørensen, M. Brandbyge, K.W. Jacobsen, Mechanical deformation of atomicscale metallic contacts: structure and mechanisms, Phys. Rev. B 57 (1998) 3283–3294. [5] N.M. Ghoniem, B.N. Singh, L.Z. Sun, T.D. de la Rubia, Interaction and accumulation of glissile defect clusters near dislocations, J. Nucl. Mater. 276 (2000) 166–177. [6] K. Detemple, O. Kanert, J.Th.M. De Hosson, K.L. Murty, In situ nuclear magnetic resonance investigation of deformation-generated vacancies in aluminum, Phys. Rev. B 52 (1995) 125–133. [7] S.B. Fisher, R.J. White, K.M. Miller, Quantitative analysis of void swelling in pure
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368
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Strain-induced transformation between vacancy voids and stacking fault tetrahedra in Cu
T
Hongbo Xva, Jie Zhaoa, , Fei Yea,b, , Ke Tonga ⁎
a b
⁎
School of Materials Science and Engineering, Dalian University of Technology, 2 Linggong Road, Dalian 116024, Liaoning, China Department of Materials Science and Engineering, Southern University of Science and Technology, No. 1088, Xueyuan Rd, Shenzhen 518055, Guangdong, China
ARTICLE INFO
ABSTRACT
Keywords: Molecular dynamics Atomistic simulation Vacancy cluster Stacking fault tetrahedra Volumetric strain
The transformation between voids and stacking fault tetrahedra (SFTs) under volumetric strain has been systematically investigated by atomistic computer simulation. It has been found out that equiaxial voids or SFTs have high stability under tensile or compressive strain. Molecular dynamic simulations show that the SFTs can transform to voids even at extremely low temperature under constant tensile strain. On the contrary, the equiaxial voids can transform to SFTs under constant compressive strain. The voids and SFTs can also transform to each other under cycling strain. During the transformation from SFTs to voids, four atoms at the vertex of SFT move outward first. Then, the {1 1 1} plane atoms move by layers toward one of the vertices. Finally, the opposing {1 1 1} plane atoms move toward the opposite direction, thereby forming planar voids. The planar voids transform to SFT through the opposite process. This strain-induced transformation is remarkably affected by temperature and strain rate. High temperature can advance the strain of the transformation, while the strain rate exerts opposite effects.
1. Introduction Various defects, including vacancies, self-interstitials, and intersecting stacking faults, are created in materials during irradiation or high-speed deformation [1–3]. Vacancies are predominant point defects, whereas the others anneal out again during the subsequent atomic rearrangements [4–6]. When a supersaturation of vacancies is induced, the vacancies profoundly influence the material properties [2,7–11]. Intensive studies have suggested that vacancies can aggregate to form voids, dislocation loops, and stacking faults [12–16]. Considerable research has been conducted to elucidate the structure of the vacancy cluster. Voids and stacking fault tetrahedra (SFTs) are dominant vacancy clusters in face-centered cubic (fcc) metals [1–3,17]. Voids can transform into SFTs by collapsing into the Frank loops (a disk of vacancies into the {1 1 1} plane), and the Frank loops then dissociate into SFTs [18,19]. Computer simulations have shown that voids collapse into the Frank loops at 800 K [20], and SFTs form around a void at 9000 K [21]. Other mechanisms have also been proposed [22]. Uberuaga showed that voids can transform directly to SFTs without passing through a Frank loop [23]. This transformation proceeds stepwise through a series of 3D structures even at a temperature as low as 400 K. Apart from temperature, stress and strain also play crucial roles in
the formation and growth of vacancy clusters [24–28]. Complicated behavior of the binding energy with uniaxial tensile strain is observed under uniaxial strain [24], and the planar cluster and void become the dominant types under high strain [29]. Under volumetric strain, the formation energy of monovacancies and divacancies monotonously decreases with the decrease in strain in Al [30]. Peng et al. [24] found via molecular static simulation that the formation energy of vacancy clusters with up to 19 vacancies in Cu increases first and then decreases under volumetric tensile strain, whereas the binding energy monotonously increases with the increase in strain, thereby increasing the cluster stability. Moreover, the stability of the clusters weakens as temperature decreases. Such effects of strain and temperature on the stability of vacancy were also showed in the molecular static simulation of Ni [31]. However, most studies on volumetric strain mainly focus on vacancy voids. The effects on the stability of the clusters with other structures, especially SFTs, have not been investigated. Moreover, since the transformation from voids to SFTs consists of a volume and a shape change in vacancy clusters [23], strain may influence the transformation, and inverse transformation is possible under certain strain and temperature condition. Actually, in the previous work of Uberuaga [23], compressive stress may occur as temperature increases under constant volume during simulation.
⁎ Corresponding authors at: School of Materials Science and Engineering, Dalian University of Technology, 2 Linggong Road, Dalian 116024, Liaoning, China (J. Zhao). E-mail addresses: [email protected] (J. Zhao), [email protected] (F. Ye).
https://doi.org/10.1016/j.commatsci.2018.11.026 Received 2 August 2018; Received in revised form 11 November 2018; Accepted 12 November 2018 0927-0256/ © 2018 Elsevier B.V. All rights reserved.
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In the present study, the effects of volumetric strain on the stability of vacancy voids and SFTs in Cu and the transformation between them are systematically studied by atomistic computer simulation. The relative stability of voids and SFTs under volumetric strain is examined by molecular static simulation. Then, the transformation between voids and SFTs is studied under constant and continuously varied strains by molecular dynamic simulation. The mechanism of the transformation under strain is also discussed.
total energy of the supercell with (N–n) atoms and an n-vacancy cluster, and ENtot is the total energy of the corresponding defect-free supercell containing N atoms under the strain. To compare the stability of the clusters, the binding energy per vacancy for an n-vacancy cluster under a strain was given by b Env =
where
F Env
N n
n N
ENtot
is the formation energy for an n-vacancy cluster,
Envf
(2)
where is the formation energy of an isolated vacancy and is the sum of E1fv . A positive binding energy indicates a preference for the n isolated vacancies to form an n-vacancy cluster, and higher binding energy denotes greater stability. The molecular dynamic calculation was performed under constant and continuously varied strains. The calculations under constant strain were performed at zero, −5%, 4%, and 5% strains. The strain higher than 4% (i.e., 5%) may result in unstable structure of the 15 V and 21 V clusters. The calculations were conducted at 10 K with additional simulation for 6 V at 1 K under zero strain. Simulation was conducted for 6 ps and sampled every 0.01 ps. The calculations under varied strains were performed from −5% to 5% for 6 V and from −5% to 4% for 15 V and 21 V. The temperatures were 300, 400, 500, and 600 K. 300 K was about room temperature, and 600 K was about the recrystallization temperature for Cu. Three strain rates, 1 × 107 s−1, 1 × 108 s−1, and 1 × 109 s−1, were applied.
The atomistic simulation based on empirical interatomic potentials was used in this work. The atomic interaction of Cu was described by an embedded atom model developed by Minish et al. [32]. This model accurately reproduces the atomic structure and energetics for vacancies in Cu [33–37]. The vacancy clusters in bulk fcc Cu were studied using classic molecular static and dynamic approaches implemented in the LAMMPS codes [38]. A cubic supercell containing 4000 (10 × 10 × 10) atoms with periodic boundary conditions along three directions was produced. The supercell was oriented in the cubic directions, that is, x in [1 0 0], y in [0 1 0], and z in [0 0 1]. The variation in the total energy of the system containing the largest cluster as a function of the supercell size was checked to ensure the size was sufficiently large. The vacancy clusters were constructed by removing atoms from the supercells. Specifically, the structure for perfect SFTs, which correspond to atoms surrounded by four {1 1 1} planes forming a regular tetrahedron, was produced by removing an equilateral triangle of atoms from one {1 1 1} plane and forming a planar vacancy cluster. Other atoms of tetrahedron above the planar cluster were collapsed to it along the 〈1 1 1〉/4 direction. Thus, the vacancy numbers of the SFTs were counted from the planar vacancy clusters. Corresponding to the formation of prefect SFTs, the vacancy clusters containing 6, 15, and 21 vacancies (denoted as 6 V, 15 V, and 21 V, respectively) were studied in this work. The most stable configurations of vacancy voids with these vacancy numbers as demonstrated in previous studies [24] are shown in Fig. 1. The 6 V void is octahedron clusters. The larger voids with 15 V and 21 V are considered as combinations of tetrahedral and octahedral clusters [29], and they appear as sphere-like shape [24]. The volumetric strain was made isotropically by applying an equal strain along the x, y, and z directions. In the molecular static calculation, the strain was ranged from −8% (compressive strain) to 8% (tensile strain) with intervals of 0.2%. To describe the energy cost for the formation of a vacancy cluster with respect to the perfect lattice under a strain, the formation energy per vacancy for an n-vacancy cluster was defined as F Env 1 tot = EnvN n n
F Env ) = E1fv
E1fv
2. Methodology
Envf =
1 ( E1fv n
E1fv
3. Results 3.1. Stability of vacancy voids and SFTs under volumetric strain The formation energy of monovacancy under volumetric strain is the basis for the calculation of binding energy. The monovacancy formation energy under zero strain is 1.3091 eV, which agrees very well with previous calculations [24]. Fig. 2 shows that the formation energy increases monotonically with the increase in volumetric strain, which agrees well with previous calculations [29,30] and confirms the validity of the calculation. The stability of the vacancy voids and SFTs was studied to compare their relative stability under strain. Figs. 3 and 4 show the average formation and binding energies under volumetric strain, respectively. As shown in Fig. 3, the formation energies increase monotonically as functions of strain for voids and SFTs consisting of 6–21 V. The increase in the formation energy for the SFTs is faster than that for voids, indicating that the strain exerts more effect on SFT than on void. The voids show lower formation energy under tensile strain, whereas the SFTs show lower formation energy under compressive strain. Thus, the voids or SFTs have higher formation tendency under tensile or compressive strain. The difference in the formation energy at certain
(1) tot Env N n
is the
Fig. 1. Structure of voids with (a) 6 V, (b) 15 V, and (c) 21 V.
Fig. 2. Formation energy for monovacancy as functions of volumetric strain. 360
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Fig. 3. Formation energy per vacancy for voids and SFTs with (a) 6 V, (b) 15 V, and (c) 21 V as functions of volumetric strain.
Fig. 4. Binding energy per vacancy for voids and SFTs with (a) 6 V, (b) 15 V, and (c) 21 V as functions of volumetric strain.
compressive or tensile strain is larger for larger clusters. Moreover, the strain at energy equivalent points in the figure, which are the intersections of the curves of voids and SFTs, are not at zero strain but under compressive strain. As shown in Table 1, the strain at energy equivalent point shifts to compressive strain for larger clusters, indicating that the voids have high formation tendency even under zero strain. In particular, the strain range of the void formation becomes larger as the vacancy number increases. The results of the binding energies are shown in Fig. 4. The binding energies of SFTs decrease with the increase in strain. On the contrary, those for voids increase with the increase in strain. Moreover, the binding energies decrease rapidly when the strain is larger than about
Table 1 Strain at energy equivalent point for 6 V, 15 V, and 21 V voids and SFTs. Vacancy number
6
15
21
Strain (%)
−3.3
−3.9
−4.1
5%. Correspondingly, the supercell structure gradually becomes disordered when the strain is further increased. As mentioned above, the structure of the 15 V and 21 V clusters becomes unstable when the strain is higher than about 4%. The strains at which the binding energies of voids and SFTs are equal are also not at zero strain, which are numerically the same as those of formation energies. 361
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The voids show lower formation energy and higher binding energy under tensile strain, whereas the SFTs show lower formation energy and higher binding energy under compressive strain. Therefore, the voids or SFTs have high stability under tensile or compressive strain. 3.2. Transformation between voids and SFTs under constant strain Since the voids or SFTs have higher formation tendency and stability under tensile or compressive strain, the voids and SFTs may transform between each other under certain strain condition. Therefore, molecular dynamic simulation was then performed under zero, −5%, 4%, and 5% strains, respectively. The results show that the transformation takes place very fast after starting the simulation even under very low temperature. For instance, under zero strain and 1 K, the transformation of 6 V SFT to void only takes 0.3 ps. To compare the effect of strain on structure transformation, the following calculations were conducted at a relatively low temperature of 10 K. The results in Fig. 5 show that most vacancy clusters can transform to a relatively stable structure as expected. Specifically, under −5% strain, the voids transform into SFTs. By contrast, under zero, 4%, and 5% strains, the SFTs transform into voids. The accomplishment of the transformation was determined from the geometries of the cluster structure observed in the visualization software OVITO [39]. The details of the transformation from SFT to void are affected by the cluster size and strain, as shown in Fig. 5. The transformations from the 6 V SFT to void can take place under zero, 4%, and 5% strains (see supplementary video S1). While the 15 V and 21 V SFTs cannot transform to voids under zero strain. When strain increases to 4%, the 15 V and 21 V SFTs transform to planar voids on the {1 1 1} plane (see supplementary videos S2 and S3). As strain increases to 5%, the voids transformed from the 15 V and 21 V SFTs expand gradually, thereby leading to the disorderliness of the supercell structure. Furthermore, the transformation time shows dependence on cluster size. The 6 V SFTs at zero, 4%, and 5% strains take around 0.4 ps to finish transforming into voids. The transformations from the 15 V and 21 V SFTs to voids at 4% strain take less than 4 ps.
supplementary video S2. Transformation from 15 V SFT to void at 10 K under constant strain 4%.
supplementary video S3. Transformation from 21 V SFT to void at 10 K under constant strain 4%.
The atomic scale processes of the transformation from SFT to void are shown in Fig. 6. In the case of 15 V and 21 V clusters (Fig. 6(c)–(f)), the figure shows that four atoms at the vertex of SFT move outward along 〈1 1 1〉 direction first. Then, the atoms on {1 1 1} planes covered by pink polyhedra move by layers toward one of the vertices along [11¯1] direction. Finally, the atoms on the opposing {1 1 1} plane marked by blue surface move along the opposite [1¯11¯] direction. At the end of the migration, a planar void on the {1 1 1} plane is formed (see supplementary videos S4 and S5). The transformation of SFT with 6 V is completed by expanding the four atoms outward along 〈1 1 1〉 direction because only four atoms are present in the center of this SFT (Fig. 6(a) and (b)). supplementary video S1. Transformation from 6 V SFT to void at 10 K under constant strain 4%.
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this range, the structure of the vacancy cluster can compete between void and SFT. Specifically, no such structure competition is observed for 15 V. Fig. 7 also shows that the strain of the start transformation surface increases with the increase in temperature. Thus, the transformation can take place under large strain because the temperature can promote the mobility of the atoms. The strain of the start transformation surface also increases with the decrease in strain rate because low strain rate provides sufficient time for atoms to migrate. However, the strain of finish transformation surface does not show such correlation with temperature and strain rate. The structures before and after the transformation are shown in Fig. 8. The figure shows that equiaxial voids with 6 V, 15 V, and 21 V can transform to perfect SFTs. The initial structure of the voids can maintain its configuration before transformation even under high temperature (see supplementary videos S6 and S7).
supplementary video S4. Atomic details of transformation from 15 V SFT to void at 10K under constant strain 4%.
supplementary video S6. Transformation from 15 V void to SFT at 400 K with 1 × 108 s−1 strain rate from tensile to compressive strain.
supplementary video S5. Atomic details of transformation from 21 V SFT to void at 10K under constant strain 4%.
In previous results of voids transforming into SFTs [23], the transformation occurs at relatively high temperature (higher than 400 K) at constant volume condition. The transformation in the present work takes place at much lower temperature. This difference is possibly due to the large strain in our work. The initial structure of void is equiaxial in the transformation from void to SFT. However, the structure of void transformed from SFT becomes planar in the current work. If the planar void transforms to SFT under compressive strain, then its specific process should be the opposite, as that shown in Fig. 6. This will be further verified by the results of cyclic strain. 3.3. Transformation between voids and SFTs under varied strains The transformation between void and SFT can also takes place under varied strains and is affected by temperature and strain rate. Fig. 7 shows the strains of structure transformation from tensile to compressive strain under different temperatures and strain rates connected into surfaces. In this transformation process, the upper and nether surfaces are start and finish surface for the structure transformation, respectively. As shown in the figure, a strain range between the two surfaces exists in the structure transformation for 6 V and 21 V. In
supplementary video S7. Transformation from 21 V void to SFT at 300 K with 1 × 108 s−1 strain rate from tensile to compressive strain.
Fig. 9 shows the transformation surfaces from compressive to tensile strain under different temperatures and strain rates. In contrast to Fig. 7, the nether surface is the start transformation surface. As shown in the figure, the structures for 15 V and 21 V clusters keep stable after transformation, whereas that for 6 V cluster still shows structure
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Fig. 5. Configurations of vacancy clusters before and after the transformation between voids and SFTs under (a) −5%, (b) 0%, (c) 4%, and (d) 5% constant strains. Atoms are colored on the basis of centrosymmetry parameter (CSP) [9,40]. Atoms with CSP values smaller than 3 are removed for a better view. The same method is also used in Fig. 6(a), (c), and (e) and Figs. 8, 10, and 12. See supplementary video S1, 2 and 3 for details. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
competition in a certain strain range. In addition, the strain of the start transformation surface increases with the decrease in temperature and increases with the increase in strain rate. In this transformation process, the effects of temperature and strain rate are the same as in the transformation from tensile to compressive strain. The structures before and after the transformation of 6 V, 15 V, and 21 V clusters are shown in Fig. 10. The figure shows that the 6 V SFT can transform to equiaxial void, while the 15 V and 21 V SFTs can transform to planar voids on the {1 1 1} plane. Since void and SFT can transform to each other under varied strains, they may transform cyclically under cyclic strain condition. This cyclical transformation is observed when the strain is from −5% to 4% and immediately back to −5% under different temperatures and strain rates (see supplementary videos S8 and S9). The transformations from −5% to 4% are the same as those shown in Figs. 9 and 10. The results of the transformations from 4% to −5% are shown in Figs. 11 and 12. Notably, the initial structure of void is planar in the transformation from void to SFT. As shown in Fig. 11, the transformation surfaces for 15 V and 21 V clusters show no correlation with temperature and strain rate, which is because there are differences between the structures of planar voids transformed from SFTs in the first step.
supplementary video S9. Transformation from 21 V SFT to void and then back to SFT at 400 K with 1 × 108 s−1 strain rate under cycle strain.
4. Conclusions The transformation between voids and SFTs under volumetric strain are systematically investigated by computer simulation. The following conclusions have been obtained: (1) Molecular static calculation shows that equiaxial voids or SFTs have high stability under tensile or compressive strain. Molecular dynamic simulations show that SFTs can transform to voids even at an extremely low temperature of 1 K under constant tensile strain. On the contrary, equiaxial voids can transform to SFTs under constant compressive strain. The transformations take place very fast. For instance, the transformation from 15 V SFT to void at 4% under 10 K only takes 4 ps after starting the simulation. (2) The atomic mechanism of transformation from SFT to void is observed. First, four atoms at the vertex of SFT move outward. Then, the {1 1 1} plane atoms move by layers toward one of the vertices. Finally, the opposing {1 1 1} plane atoms move toward the opposite direction. If the planar void transforms to SFT under compressive strain, then its specific process is opposite.
supplementary video S8. Transformation from 15 V SFT to void and then back to SFT at 400 K with 1 × 108 s−1 strain rate under cycle strain. 364
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Fig. 6. The detailed structures of the clusters for the transformation from (a) 6 V, (c) 15 V, and (e) 21 V SFT to void under 10 K, and their corresponding abstract structures (b), (d), and (f), respectively. The red and blue spheres in the model structure represent the atoms at lattice point and tetrahedral interstices, respectively. The surface in (d) and (f) marks the atoms which will migrate in 〈1 1 1〉 direction. See supplementary videos S4 and S5 for details.
(3) Voids and SFTs can transform to each other under varied strains and can also transform cyclically under cyclic strains. In these processes, temperature and strain rate play important roles. High temperature can ease structure transformation by increasing the mobility of atoms. Meanwhile, low strain rate can provide plenty of
time for the migration of atoms. Consequently, higher temperature and lower strain rate can promote the transformation and advance the strain of transformation.
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Fig. 7. Transformation surfaces at which the equiaxial voids with (a) 6 V, (b) 15 V, and (c) 21 V transform to SFTs from tensile to compressive strain.
Fig. 10. Structures of vacancy cluster before and after the transformation from SFTs with (a) 6 V, (b) 15 V, and (c) 21 V to voids when the strain varied from compression to tension. The voids transformed from SFTs with 15 V and 21 V are on the {1 1 1} planes.
Fig. 8. Structures of vacancy cluster before and after the transformation from equiaxial voids with (a) 6 V, (b) 15 V, and (c) 21 V to SFTs when the strain varied from tension to compression. See supplementary videos S6, S7 for details.
Fig. 9. Transformation surfaces at which the SFTs with (a) 6 V, (b) 15 V, and (c) 21 V transform to {1 1 1} plane voids from compressive to tensile strain.
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Fig. 12. Structures of vacancy cluster before and after the transformation from voids with (a) 6 V, (b) 15 V, and (c) 21 V to SFTs under cycle strain. The structure of voids is the same as that shown in Fig. 10. See supplementary videos S8, S9 for details.
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