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The energy and structure of (1 1 0) twist grain boundary in tungsten
Applied Surface Science 357 (2015) 262–267
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
The energy and structure of (1 1 0) twist grain boundary in tungsten Ya-Xin Feng a , Jia-Xiang Shang a,∗ , Zeng-Hui Liu a , Guang-Hong Lu b a b
School of Materials Science and Engineering, Beihang University, Beijing 100191, China School of Physics and Nuclear Engineering, Beihang University, Beijing 100191, China
a r t i c l e
i n f o
Article history: Received 1 July 2015 Received in revised form 22 August 2015 Accepted 31 August 2015 Available online 3 September 2015 Keywords: Twist grain boundaries Tungsten Molecular statics Dislocation
a b s t r a c t The energy and structure of (1 1 0) twist grain boundary (TGB) in W are systematically investigated using molecular statics simulations. A dependence of grain boundary energy vs. the twist angle over the entire angular range is obtained and agrees well with the modified Read–Shockley equation in low-angle range. According to the analysis of dislocations in grain boundaries, (1 1 0) TGBs can be divided into three types: low-angle grain boundaries (LAGBs), intermediate-angle grain boundaries (IAGBs) and high-angle grain boundaries (HAGBs). When ≤ 16.10◦ , a regular dislocation network consisting of 12 [1 1 1], 12 [1¯ 1¯ 1] and [0 0 1] screw dislocations exists in LAGBs. And the size and shape of meshes in the network vary with increased twist angles. In IAGBs, with 17.23◦ ≤ ≤ 22.22◦ , both atomistic structures and dislocations are disordered and dislocations do not form a regular network. The TGBs with ≥ 23.5◦ are HAGBs, where no dislocation is observed. The HAGBs can be divided into three sub-types further: special boundaries with low ˙, boundaries in their vicinity with similar structures as the corresponding special boundary in local regions as well as ordinary HAGBs consisting of periodic patterns. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Grain boundaries have a significant influence on the properties of polycrystalline materials, of which twist grain boundaries (TGBs) receive considerable attention due to their unique grain boundary structures. A number of attempts have been made to study the energy and structure of TGBs, especially for face-centered cubic materials [1–14]. Through atomistic simulation and the generalized Peierls–Nabarro model, Dai et al. [13,14] study the energy and structure of (1 1 1) TGBs in face-centered cubic Al, Cu and Ni for all twist angles. They classify all (1 1 1) TGBs into three types: low-angle grain boundaries (LAGBs), near-twin grain boundaries (NTGBs) and those intermediate between the two (IAGBs). Dislocation networks made up by partial dislocations are observed in LAGBs and NTGBs. And the regions separated by partial dislocations are perfect crystal and stacking fault in LAGBs but twin boundaries in NTGBs. The structures of IAGBs are disordered. Besides, analytical expressions for (1 1 1) twist grain boundary energy as a function of twist angle over the entire twist angle range are proposed. However, for body-centered cubic (BCC) materials, researches about TGB are mainly focused on the low-angle boundaries with a hexagonal dislocation network [15–17]. A comprehensive understanding
∗ Corresponding author. E-mail address: [email protected] (J.-X. Shang). http://dx.doi.org/10.1016/j.apsusc.2015.08.265 0169-4332/© 2015 Elsevier B.V. All rights reserved.
of the energy and structure of TGBs over an entire twist angle range in BCC materials is lacking.
2. Method We perform the molecular statics (MS) simulation to study the energy and structure of (1 1 0) TGB in W over the entire misorientation range (0 ≤ ≤ 90◦ , due to lattice symmetry) via the widely used LAMMPS [18] code with an embedded-atom method (EAM) potential [19]. Previous researches [20–22] show that the potential can describe the interaction between W–W atoms properly. The initial configuration of our model is a single crystal of ¯ W, and its axes are x [1 1 0], y [0 0 1] and z[110]. Then we rotate the upper and lower half crystals at the midplane about z-axes by /2 clockwise and counterclockwise respectively. After rotation, the axes of upper and lower grains are x [m m n] and ¯ y[n¯ n¯ 2 m] and [n n 2 m] respectively, where m, n are pos[m m n], itive integers and are given in Table √ 1. The twist angle can be calculated through = 2 arctan(n/ 2m), and the reciprocal density of coincident sites can be calculated by ˙ = 2m2 + n2 . To insure the periodic boundary condition along all three directions, the box dimensions Lx and Ly can be adjusted to several times of the lengths of the√crystal directions along x-axes and y-axes, and Lz is fixed at 40 2a corresponding to 80 atomic layers, where the ˚ The top and bottom 10 atomic laylattice parameter a = 3.165A. ers are fixed to avoid the influence between boundaries. Static
Y.-X. Feng et al. / Applied Surface Science 357 (2015) 262–267
263
Table 1 , ˙, n and m of the calculated (1 1 0) TGBs. (◦ )
˙
n
m
(◦ )
˙
n
m
1.00 2.03 3.00 4.05 5.06 6.23 7.36 8.09 8.98 10.10 11.54 12.11 13.44 14.23 15.10 16.10 17.23 18.54 20.05 21.08 22.22 23.50 24.55 26.53 28.03 29.70 30.43 31.59 32.47 33.16 34.38 36.52 37.04 38.94 40.83 41.05 42.18 43.38 44.00 45.98 47.03
13123 3201 1459 801 513 339 243 201 163 129 99 809 73 587 521 51 401 347 33 747 673 603 177 19 307 137 363 27 627 393 561 499 803 9 411 659 139 593 57 59 201
1 1 1 1 1 1 1 1 1 1 1 3 1 3 3 1 3 3 1 5 5 5 4 1 6 3 5 2 7 8 7 7 9 1 10 9 6 9 4 3 8
81 40 27 20 16 13 11 10 9 8 7 20 6 17 16 5 14 13 4 19 18 17 13 3 17 8 13 5 17 19 16 15 19 2 19 17 11 16 7 5 13
48.13 49.17 50.48 51.64 52.17 53.05 54.11 55.88 57.09 58.11 59.76 61.02 62.02 63.00 64.30 65.47 66.27 67.08 67.92 68.92 70.53 71.50 72.15 73.32 74.05 75.29 76.31 77.00 78.42 79.04 80.63 81.30 82.07 83.70 84.55 85.18 86.63 87.91 88.39 89.42
433 699 11 891 419 361 817 41 219 513 681 97 1089 619 113 171 241 737 641 3401 3 4593 1667 561 353 193 131 209 563 121 43 681 297 811 179 369 17 467 249 99
12 11 2 13 9 12 13 3 10 11 13 5 17 13 8 10 12 15 20 33 1 56 34 20 16 12 10 9 15 7 6 17 16 19 9 13 4 15 11 7
19 17 3 19 13 17 18 4 13 14 16 6 20 15 9 11 13 16 21 34 1 55 33 19 15 11 9 8 13 6 5 14 13 15 7 10 3 11 8 5
relaxation is achieved by a conjugate gradient energy minimization. The grain boundary energy (GBE) can be calculated by =
(Ei − Ec )/A
i
where Ei , Ec is the potential energy of atoms in TGB and perfect crystal respectively and A is the area of x–y plane. The calculation is done in the middle 40 atomic layers, which aims to exclude the influence of ambilateral boundaries or fixed atoms. AtomEye [23] and dislocation extraction algorithm (DXA) [24] are used to analysis the atomistic structure and dislocation of TGB. 3. Results and discussion 3.1. Grain boundary energy Fig. 1 indicates the dependence of GBE vs. the twist angle over the entire angular range. As Fig. 1 shows, the GBE rises rapidly and approaches an asymptotic value at low angles. At higher angles, some cusps corresponding to the energies of the relatively low ˙ boundaries are formed. The GBE curve has the similar form as that of (1 1 0) TGB in ˛-iron [15], which indicates this form may be universal to BCC metals. In order to describe the misorientation dependence of the GBE in low-angle tilt boundaries consisting of an array of parallel edge dislocations, the Read–Shockley equation [25] is proposed as = 0 (A − ln ), where both 0 and A are constants
Fig. 1. The grain boundary energy (GBE) as a function of the twist angle . The red line is the fitted curve based on the improved Read–Shockley equation for ≤ 16.10◦ . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
and is misorientation. However, the TGB contains a crossed grid of screw dislocations, Vitek [26] adds an additional term representing the interaction between the boundary and crossed dislocations into the Read–Shockley equation, = 0 (A + B − ln ), where B is another constant. The improved equation is used to fit the simulation data for ≤ 16.10◦ with 0 = 0.199 J/m2 , A = 2.768 and B = 0.036. The fitted curve agrees well with the simulation data for ≤ 16.10◦ and deviates at higher angles. As there is no regular dislocation network in TGBs for ≥ 16.10◦ , the improved Read–Shockley equation based on dislocation model does not work. 3.2. Grain boundary structure According to the analysis of dislocations, TGBs can be divided into three types: low-angle grain boundaries (LAGBs), intermediate-angle grain boundaries (IAGBs) and high-angle grain boundaries (HAGBs), corresponding to different stages of the GBE curve respectively. When ≤ 16.10◦ , a dislocation network exists in LAGBs, corresponding to the rapid rise stage of the GBE. Before relaxation, the crossed 12 1 1 1 screw dislocations make up a periodic tetragonal dislocation network. And the network transforms into a hexagonal dislocation network by the dislocation reaction: 12 [1 1 1] + 1 ¯ ¯ [1 1 1] = [0 0 1] during relaxation. In this reaction, both parents 2 and junction dislocations are pure screw dislocations, which is often observed for BCC metals [27,28]. Fig. 2 shows the structure variation of relaxed LAGBs along with increased twist angles, where Fig. 2a–c are atomistic structures with different twist angles and Fig. 2d–f are corresponding dislocation networks. As Fig. 2a and d show, a hexagonal dislocation network consisting of 12 [1 1 1], 1 ¯ ¯ [1 1 1] and [0 0 1] screw dislocations exists in (1 1 0) TGB for 2 = 2.03◦ . In experiment, Ryan et al. [29] observe screw dislocations in the (1 1 0) TGB of W with a misorientation of approximately 6◦ , which agrees with our simulation results. With the increase of the twist angle, the size of meshes of the dislocation network decreases and some hexagonal meshes change into tetragonal meshes for ≥ 8.98◦ gradually (Fig. 2a–f). Fig. 2g and h show the displacements of upper and lower layer atoms in TGB with = 2.03◦ . As seen in Fig. 2g and h, the dislocation network divides the TGB into separated regions and atoms of each region rotate around its center clockwise and counterclockwise respectively, contrary to the rotation direction when the model is set up. The magnitude of displacements increases gradually from center to edge in each region. According
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Fig. 2. (a–f) Atomistic structure and dislocations of LAGBs with three twist angles: (a) and (d) = 2.03◦ , (b) and (e) = 8.98◦ , (c) and (f) = 16.10◦ . Atoms and dislocation lines are colored according to the potential energy and the magnitude of the Burgers vectors respectively, as indicated by the color bars. Arrows in (d–f) represent the directions of Burgers vectors. (g and h) The disregistry plot of the upper (lower) layer atoms in TGB with = 2.03◦ . The length and color of arrows indicate the magnitude of the displacements, as indicated by the color bar. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
to the theory of O-lattice [30], the O-points, corresponding to the centers of separated regions, are the fixed points and origins for the relaxation in TGB. The upper and lower grain rotate with respect to one another around each O-point to restore perfect lattice matching across each local region. The relaxation starts simultaneously from all O-points until it comes across cell walls (dislocations). The spacing of the O-points decreases monotonically with the increased twist angle. This process improves lattice matching across TGB with a reduction of the GBE. When 17.23◦ ≤ ≤ 22.22◦ , both atomistic structures and the structures of dislocations are disordered in IAGBs, as shown in Fig. 3. Besides, the dislocations change from screw dislocations into mixed dislocations and their Burgers vectors vary correspondingly
(Fig. 3d–f). In this stage, the dislocation cores approach one another and finally impinge [31]. The interaction between dislocation cores may be the cause for this disorder. Accompanied with this process, the GBE rises slowly and reaches a stable value eventually. When ≥ 23.50◦ , no dislocation is detected by DXA in HAGBs. The TGBs consist of periodic patterns, as shown in Fig. 4a–f. According to the atomistic structures and corresponding GBEs, HAGBs can be divided into three sub-types further: special boundaries (Fig. 4a), boundaries in the vicinity of special boundaries (Fig. 4b) and ordinary HAGBs (Fig. 4c). Special boundaries are boundaries with relatively low ˙ and their GBEs correspond to the cusps of the GBE curve. Due to their low ˙s, special boundaries have good symmetry and lattice matching. Before relaxation, the structures
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Fig. 3. (a–c) Atomistic structure and dislocations of IAGBs with three twist angles: (a) and (d) = 17.23◦ , (b) and (e) = 18.54◦ , (c) and (f) = 21.08◦ . Atoms and dislocation lines are colored according to the potential energy and the magnitude of the Burgers vectors respectively, as indicated by the color bars. Arrows in (d–f) represent the directions of Burgers vectors. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Atomistic structure of HAGBs with different twist angles: (a) = 50.48◦ , (b) = 48.13◦ , (c) = 83.70◦ , (d) = 65.47◦ , (e) = 67.08◦ , (f) = 70.53◦ . Atoms are colored according to the potential energy, as indicated by the color bar. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 5. The disregistry plot of the upper (lower) layer atoms in special boundaries: (a) and (b) = 26.53◦ , (c) and (d) = 31.59◦ , (e) and (f) = 38.94◦ , (g) and (h) = 50.48◦ , (i) and (j) = 70.53◦ . The length and color of arrows indicate the magnitude of the displacements, as indicated by the color bar. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Y.-X. Feng et al. / Applied Surface Science 357 (2015) 262–267
of special boundaries are relatively stable. And they reach stable states with little atomic displacements during relaxation (Fig. 5). As a result, special boundaries tend to have relatively low GBEs with an exception for ˙19 boundary, which is explained below. Boundaries in the vicinity of special boundaries have similar structures as the corresponding special boundary in local regions which improves their matching and reduce their GBEs. Fig. 4d–f are atomistic structures of the ˙3 TGB and boundaries in its vicinity. Combined with Fig. 1, we find that the atomistic structure gets close to that of ˙3 TGB gradually and the GBE decreases correspondingly as the approach of misorientation. The ordinary HAGBs are not influenced by special boundaries. They consist of periodic patterns and their GBEs near a stable value. Fig. 5 shows the displacements of upper and lower layer atoms of special boundaries during relaxation. As shown in Fig. 5, periodic rows or columns consisting of atoms which move towards each other are observed. And the positions of these rows or columns in upper and lower layers are interleaved with each other. According to these observations, we can make some conclusions. Before relaxation, some atoms in the upper and lower layers coincide with each other or close to each other due to the good symmetry of special boundaries. The strong repulsive force makes them move away from each other which reduce their GBEs. However, in ˙19 TGB, some atoms in the same layer come too close to each other after relaxation which can be seen in Fig. 5a and b where the red and yellow arrows point to almost same positions. The repulsion between these atoms in the same layer increases the GBE of ˙19 TGB. In other special boundaries (Fig. 5c–j), atoms in the same layer do not come so close and produce strong repulsion. Therefore, their GBEs are relatively low. It’s worth noting that the GBE of ˙19 boundary after relaxation is much lower than that of ˙19 boundary before relaxation even through there is repulsion. 4. Conclusion In summary, we study the energy and structure of (1 1 0) TGB in W over the full range of twist angles through molecular statics simulation. A dependence of GBE vs. the twist angle is obtained and agrees well with the improved Read–Shockley equation in lowangle range. The TGBs can be divided into three types: low-angle grain boundaries (LAGBs), with twist angle ≤ 16.10◦ , contain an intact dislocation network consisting of 12 [1 1 1], 21 [1¯ 1¯ 1] and [0 0 1] screw dislocations, which coincides with observations in experiment. The TGBs with 17.23◦ ≤ ≤ 22.22◦ are intermediate-angle grain boundaries (IAGBs), where both atomistic structures and dislocations are disordered and dislocations do not form a regular network. The TGBs with ≥ 23.5◦ are high-angle grain boundaries (HAGBs), where no dislocation is observed. The HAGBs can be divided into three sub-types further: special boundaries with low ˙, boundaries in their vicinity with similar structures as the corresponding special boundary in local regions as well as ordinary HAGBs consisting of periodic patterns. Acknowledgement This work is supported by the National Magnetic Confinement Fusion Program through Grant No. 2013GB109002.
267
References [1] X.M. Wei, J.M. Zhang, K.W. Xu, The energy and structure of (0 0 1) twist grain boundary in noble metals, Appl. Surf. Sci. 253 (2006) 854–858. [2] X.M. Wei, J.M. Zhang, K.W. Xu, The periodicity in translation of Ag (0 0 1) and (1 1 0) twist grain boundary, Appl. Surf. Sci. 253 (2007) 4307–4310. [3] X.M. Wei, J.M. Zhang, K.W. Xu, Energy calculation of (0 1 1) twist grain boundary in noble metals, Appl. Surf. Sci. 252 (2006) 7331–7336. [4] J.M. Zhang, X.M. Wei, H. Xin, Energy analysis for (1 1 1) twist grain boundary in noble metals, Appl. Surf. Sci. 243 (2005) 1–6. [5] N.A. Gjostein, F.N. Rhines, Absolute interfacial energies of [0 0 1] tilt and twist grain boundaries in copper, Acta Metall. 7 (1959) 319–330. [6] A. Seki, O. Hellman, S. Tanaka, Calculation of grain boundary energies and structures in copper [0 0 1] twist boundaries using the modified embedded atom method, Scripta Mater. 34 (1996) 1867–1870. [7] D. Wolf, Effect of interatomic potential on the calculated energy and structure of high-angle coincident site grain boundaries – II. (1 0 0) twist boundaries in Cu, Ag and Au, Acta Metall. 32 (1984) 735–748. [8] A. Otsuki, Energies of (0 0 1) twist grain boundaries in silicon, Acta Mater. 49 (2001) 1737–1745. [9] M.R. Fitzsimmons, S.L. Sass, Quantitative X-ray diffraction study of the atomic structure of the ˙ = 5( = 36.9◦ ) [0 0 1] twist boundary in gold, Acta Metall. 36 (1988) 3103–3122. [10] P.D. Bristowe, S.L. Sass, The atomic structure of a large angle [0 0 1] twist boundary in gold determined by a joint computer modelling and X-ray diffraction study, Acta Metall. 28 (1980) 575–588. [11] M.R. Fitzsimmons, S.L. Sass, The atomic structure of the ˙ = 13( = 22.6◦ ) [0 0 1] twist boundary in gold determined using quantitative X-ray diffraction techniques, Acta Metall. 37 (1989) 1009–1022. [12] D. Wolf, Effect of interatomic potential on the calculated energy and structure of high-angle coincident site grain boundaries – I. (1 0 0) twist boundaries in aluminum, Acta Metall. 32 (1984) 245–258. [13] S. Dai, Y. Xiang, D.J. Srolovitz, Atomistic, generalized Peierls–Nabarro and analytical models for (1 1 1) twist boundaries in Al, Cu and Ni for all twist angles, Acta Mater. 69 (2014) 162–174. [14] S. Dai, Y. Xiang, D.J. Srolovitz, Structure and energy of (1 1 1) low-angle twist boundaries in Al, Cu and Ni, Acta Mater. 61 (2013) 1327–1337. [15] J.B. Yang, Y. Nagai, M. Hasegawa, Yu.N. Osetsky, Atomic scale modeling of {1 1 0} twist grain boundaries in ˛-iron: structure and energy properties, Philos. Mag. 90 (2010) 991–1000. [16] J.B. Yang, Y.N. Osetsky, R.E. Stoller, Y. Nagai, M. Hasegawa, The effect of twist angle on anisotropic mobility of {1 1 0} hexagonal dislocation networks in ˛-iron, Scripta Mater. 66 (2012) 761–764. [17] V.V. Bulatov, W. Cai, Nodal effects in dislocation mobility, Phys. Rev. Lett. 89 (2002) 115501. [18] S.J. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys. 117 (1995) 1–19. [19] N. Juslin, B.D. Wirth, Interatomic potentials for simulation of He bubble formation in W, J. Nucl. Mater. 432 (2013) 61–66. [20] A.V. Barashev, H. Xu, R.E. Stoller, The behavior of small helium clusters near free surfaces in tungsten, J. Nucl. Mater. 454 (2014) 421–426. [21] L. Hu, K.D. Hammond, B.D. Wirth, D. Maroudas, Dynamics of small mobile helium clusters near tungsten surfaces, Surf. Sci. 626 (2014) L21–L25. [22] F. Sefta, N. Juslin, K.D. Hammond, B.D. Wirth, Molecular dynamics simulations on the effect of sub-surface helium bubbles on the sputtering yield of tungsten, J. Nucl. Mater. 438 (2013) S493–S496. [23] J. Li, AtomEye: an efficient atomistic configuration viewer, Modell. Simul. Mater. Sci. Eng. 11 (2003) 173–177. [24] A. Stukowski, K. Albe, Extracting dislocations and non-dislocation crystal defects from atomistic simulation data, Modell. Simul. Mater. Sci. Eng. 18 (2010) 085001. [25] W.T. Read, W. Shockley, Dislocation models of crystal grain boundaries, Phys. Rev. 78 (1950) 275–289. [26] V. Vitek, On the difference between the misorientation dependences of the energies of tilt and twist boundaries, Scripta Metall. 21 (1987) 711–714. [27] S.M. Ohr, D.N. Beshers, Crystallography of dislocation networks in annealed iron, Philos. Mag. 8 (1963) 1343–1360. [28] A.J. Garratt-reed, G. Taylor, Optical and electron microscopy of niobium crystals deformed below room temperature, Philos. Mag. A 39 (1979) 597–646. [29] H.F. Ryan, J. Suiter, Observations of dislocations in high angle boundaries in tungsten, Acta Metall. 14 (1966) 847–851. [30] W. Bollmann, Crystal Defects and Crystalline Interfaces, Springer, Berlin, 1970. [31] A. Brokma, R.W. Balluffi, Coincidence lattice model for the structure and energy of grain boundaries, Acta Metall. 29 (1981) 1703–1719.
Contents lists available at ScienceDirect
Applied Surface Science journal homepage: www.elsevier.com/locate/apsusc
The energy and structure of (1 1 0) twist grain boundary in tungsten Ya-Xin Feng a , Jia-Xiang Shang a,∗ , Zeng-Hui Liu a , Guang-Hong Lu b a b
School of Materials Science and Engineering, Beihang University, Beijing 100191, China School of Physics and Nuclear Engineering, Beihang University, Beijing 100191, China
a r t i c l e
i n f o
Article history: Received 1 July 2015 Received in revised form 22 August 2015 Accepted 31 August 2015 Available online 3 September 2015 Keywords: Twist grain boundaries Tungsten Molecular statics Dislocation
a b s t r a c t The energy and structure of (1 1 0) twist grain boundary (TGB) in W are systematically investigated using molecular statics simulations. A dependence of grain boundary energy vs. the twist angle over the entire angular range is obtained and agrees well with the modified Read–Shockley equation in low-angle range. According to the analysis of dislocations in grain boundaries, (1 1 0) TGBs can be divided into three types: low-angle grain boundaries (LAGBs), intermediate-angle grain boundaries (IAGBs) and high-angle grain boundaries (HAGBs). When ≤ 16.10◦ , a regular dislocation network consisting of 12 [1 1 1], 12 [1¯ 1¯ 1] and [0 0 1] screw dislocations exists in LAGBs. And the size and shape of meshes in the network vary with increased twist angles. In IAGBs, with 17.23◦ ≤ ≤ 22.22◦ , both atomistic structures and dislocations are disordered and dislocations do not form a regular network. The TGBs with ≥ 23.5◦ are HAGBs, where no dislocation is observed. The HAGBs can be divided into three sub-types further: special boundaries with low ˙, boundaries in their vicinity with similar structures as the corresponding special boundary in local regions as well as ordinary HAGBs consisting of periodic patterns. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Grain boundaries have a significant influence on the properties of polycrystalline materials, of which twist grain boundaries (TGBs) receive considerable attention due to their unique grain boundary structures. A number of attempts have been made to study the energy and structure of TGBs, especially for face-centered cubic materials [1–14]. Through atomistic simulation and the generalized Peierls–Nabarro model, Dai et al. [13,14] study the energy and structure of (1 1 1) TGBs in face-centered cubic Al, Cu and Ni for all twist angles. They classify all (1 1 1) TGBs into three types: low-angle grain boundaries (LAGBs), near-twin grain boundaries (NTGBs) and those intermediate between the two (IAGBs). Dislocation networks made up by partial dislocations are observed in LAGBs and NTGBs. And the regions separated by partial dislocations are perfect crystal and stacking fault in LAGBs but twin boundaries in NTGBs. The structures of IAGBs are disordered. Besides, analytical expressions for (1 1 1) twist grain boundary energy as a function of twist angle over the entire twist angle range are proposed. However, for body-centered cubic (BCC) materials, researches about TGB are mainly focused on the low-angle boundaries with a hexagonal dislocation network [15–17]. A comprehensive understanding
∗ Corresponding author. E-mail address: [email protected] (J.-X. Shang). http://dx.doi.org/10.1016/j.apsusc.2015.08.265 0169-4332/© 2015 Elsevier B.V. All rights reserved.
of the energy and structure of TGBs over an entire twist angle range in BCC materials is lacking.
2. Method We perform the molecular statics (MS) simulation to study the energy and structure of (1 1 0) TGB in W over the entire misorientation range (0 ≤ ≤ 90◦ , due to lattice symmetry) via the widely used LAMMPS [18] code with an embedded-atom method (EAM) potential [19]. Previous researches [20–22] show that the potential can describe the interaction between W–W atoms properly. The initial configuration of our model is a single crystal of ¯ W, and its axes are x [1 1 0], y [0 0 1] and z[110]. Then we rotate the upper and lower half crystals at the midplane about z-axes by /2 clockwise and counterclockwise respectively. After rotation, the axes of upper and lower grains are x [m m n] and ¯ y[n¯ n¯ 2 m] and [n n 2 m] respectively, where m, n are pos[m m n], itive integers and are given in Table √ 1. The twist angle can be calculated through = 2 arctan(n/ 2m), and the reciprocal density of coincident sites can be calculated by ˙ = 2m2 + n2 . To insure the periodic boundary condition along all three directions, the box dimensions Lx and Ly can be adjusted to several times of the lengths of the√crystal directions along x-axes and y-axes, and Lz is fixed at 40 2a corresponding to 80 atomic layers, where the ˚ The top and bottom 10 atomic laylattice parameter a = 3.165A. ers are fixed to avoid the influence between boundaries. Static
Y.-X. Feng et al. / Applied Surface Science 357 (2015) 262–267
263
Table 1 , ˙, n and m of the calculated (1 1 0) TGBs. (◦ )
˙
n
m
(◦ )
˙
n
m
1.00 2.03 3.00 4.05 5.06 6.23 7.36 8.09 8.98 10.10 11.54 12.11 13.44 14.23 15.10 16.10 17.23 18.54 20.05 21.08 22.22 23.50 24.55 26.53 28.03 29.70 30.43 31.59 32.47 33.16 34.38 36.52 37.04 38.94 40.83 41.05 42.18 43.38 44.00 45.98 47.03
13123 3201 1459 801 513 339 243 201 163 129 99 809 73 587 521 51 401 347 33 747 673 603 177 19 307 137 363 27 627 393 561 499 803 9 411 659 139 593 57 59 201
1 1 1 1 1 1 1 1 1 1 1 3 1 3 3 1 3 3 1 5 5 5 4 1 6 3 5 2 7 8 7 7 9 1 10 9 6 9 4 3 8
81 40 27 20 16 13 11 10 9 8 7 20 6 17 16 5 14 13 4 19 18 17 13 3 17 8 13 5 17 19 16 15 19 2 19 17 11 16 7 5 13
48.13 49.17 50.48 51.64 52.17 53.05 54.11 55.88 57.09 58.11 59.76 61.02 62.02 63.00 64.30 65.47 66.27 67.08 67.92 68.92 70.53 71.50 72.15 73.32 74.05 75.29 76.31 77.00 78.42 79.04 80.63 81.30 82.07 83.70 84.55 85.18 86.63 87.91 88.39 89.42
433 699 11 891 419 361 817 41 219 513 681 97 1089 619 113 171 241 737 641 3401 3 4593 1667 561 353 193 131 209 563 121 43 681 297 811 179 369 17 467 249 99
12 11 2 13 9 12 13 3 10 11 13 5 17 13 8 10 12 15 20 33 1 56 34 20 16 12 10 9 15 7 6 17 16 19 9 13 4 15 11 7
19 17 3 19 13 17 18 4 13 14 16 6 20 15 9 11 13 16 21 34 1 55 33 19 15 11 9 8 13 6 5 14 13 15 7 10 3 11 8 5
relaxation is achieved by a conjugate gradient energy minimization. The grain boundary energy (GBE) can be calculated by =
(Ei − Ec )/A
i
where Ei , Ec is the potential energy of atoms in TGB and perfect crystal respectively and A is the area of x–y plane. The calculation is done in the middle 40 atomic layers, which aims to exclude the influence of ambilateral boundaries or fixed atoms. AtomEye [23] and dislocation extraction algorithm (DXA) [24] are used to analysis the atomistic structure and dislocation of TGB. 3. Results and discussion 3.1. Grain boundary energy Fig. 1 indicates the dependence of GBE vs. the twist angle over the entire angular range. As Fig. 1 shows, the GBE rises rapidly and approaches an asymptotic value at low angles. At higher angles, some cusps corresponding to the energies of the relatively low ˙ boundaries are formed. The GBE curve has the similar form as that of (1 1 0) TGB in ˛-iron [15], which indicates this form may be universal to BCC metals. In order to describe the misorientation dependence of the GBE in low-angle tilt boundaries consisting of an array of parallel edge dislocations, the Read–Shockley equation [25] is proposed as = 0 (A − ln ), where both 0 and A are constants
Fig. 1. The grain boundary energy (GBE) as a function of the twist angle . The red line is the fitted curve based on the improved Read–Shockley equation for ≤ 16.10◦ . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
and is misorientation. However, the TGB contains a crossed grid of screw dislocations, Vitek [26] adds an additional term representing the interaction between the boundary and crossed dislocations into the Read–Shockley equation, = 0 (A + B − ln ), where B is another constant. The improved equation is used to fit the simulation data for ≤ 16.10◦ with 0 = 0.199 J/m2 , A = 2.768 and B = 0.036. The fitted curve agrees well with the simulation data for ≤ 16.10◦ and deviates at higher angles. As there is no regular dislocation network in TGBs for ≥ 16.10◦ , the improved Read–Shockley equation based on dislocation model does not work. 3.2. Grain boundary structure According to the analysis of dislocations, TGBs can be divided into three types: low-angle grain boundaries (LAGBs), intermediate-angle grain boundaries (IAGBs) and high-angle grain boundaries (HAGBs), corresponding to different stages of the GBE curve respectively. When ≤ 16.10◦ , a dislocation network exists in LAGBs, corresponding to the rapid rise stage of the GBE. Before relaxation, the crossed 12 1 1 1 screw dislocations make up a periodic tetragonal dislocation network. And the network transforms into a hexagonal dislocation network by the dislocation reaction: 12 [1 1 1] + 1 ¯ ¯ [1 1 1] = [0 0 1] during relaxation. In this reaction, both parents 2 and junction dislocations are pure screw dislocations, which is often observed for BCC metals [27,28]. Fig. 2 shows the structure variation of relaxed LAGBs along with increased twist angles, where Fig. 2a–c are atomistic structures with different twist angles and Fig. 2d–f are corresponding dislocation networks. As Fig. 2a and d show, a hexagonal dislocation network consisting of 12 [1 1 1], 1 ¯ ¯ [1 1 1] and [0 0 1] screw dislocations exists in (1 1 0) TGB for 2 = 2.03◦ . In experiment, Ryan et al. [29] observe screw dislocations in the (1 1 0) TGB of W with a misorientation of approximately 6◦ , which agrees with our simulation results. With the increase of the twist angle, the size of meshes of the dislocation network decreases and some hexagonal meshes change into tetragonal meshes for ≥ 8.98◦ gradually (Fig. 2a–f). Fig. 2g and h show the displacements of upper and lower layer atoms in TGB with = 2.03◦ . As seen in Fig. 2g and h, the dislocation network divides the TGB into separated regions and atoms of each region rotate around its center clockwise and counterclockwise respectively, contrary to the rotation direction when the model is set up. The magnitude of displacements increases gradually from center to edge in each region. According
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Fig. 2. (a–f) Atomistic structure and dislocations of LAGBs with three twist angles: (a) and (d) = 2.03◦ , (b) and (e) = 8.98◦ , (c) and (f) = 16.10◦ . Atoms and dislocation lines are colored according to the potential energy and the magnitude of the Burgers vectors respectively, as indicated by the color bars. Arrows in (d–f) represent the directions of Burgers vectors. (g and h) The disregistry plot of the upper (lower) layer atoms in TGB with = 2.03◦ . The length and color of arrows indicate the magnitude of the displacements, as indicated by the color bar. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
to the theory of O-lattice [30], the O-points, corresponding to the centers of separated regions, are the fixed points and origins for the relaxation in TGB. The upper and lower grain rotate with respect to one another around each O-point to restore perfect lattice matching across each local region. The relaxation starts simultaneously from all O-points until it comes across cell walls (dislocations). The spacing of the O-points decreases monotonically with the increased twist angle. This process improves lattice matching across TGB with a reduction of the GBE. When 17.23◦ ≤ ≤ 22.22◦ , both atomistic structures and the structures of dislocations are disordered in IAGBs, as shown in Fig. 3. Besides, the dislocations change from screw dislocations into mixed dislocations and their Burgers vectors vary correspondingly
(Fig. 3d–f). In this stage, the dislocation cores approach one another and finally impinge [31]. The interaction between dislocation cores may be the cause for this disorder. Accompanied with this process, the GBE rises slowly and reaches a stable value eventually. When ≥ 23.50◦ , no dislocation is detected by DXA in HAGBs. The TGBs consist of periodic patterns, as shown in Fig. 4a–f. According to the atomistic structures and corresponding GBEs, HAGBs can be divided into three sub-types further: special boundaries (Fig. 4a), boundaries in the vicinity of special boundaries (Fig. 4b) and ordinary HAGBs (Fig. 4c). Special boundaries are boundaries with relatively low ˙ and their GBEs correspond to the cusps of the GBE curve. Due to their low ˙s, special boundaries have good symmetry and lattice matching. Before relaxation, the structures
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Fig. 3. (a–c) Atomistic structure and dislocations of IAGBs with three twist angles: (a) and (d) = 17.23◦ , (b) and (e) = 18.54◦ , (c) and (f) = 21.08◦ . Atoms and dislocation lines are colored according to the potential energy and the magnitude of the Burgers vectors respectively, as indicated by the color bars. Arrows in (d–f) represent the directions of Burgers vectors. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Atomistic structure of HAGBs with different twist angles: (a) = 50.48◦ , (b) = 48.13◦ , (c) = 83.70◦ , (d) = 65.47◦ , (e) = 67.08◦ , (f) = 70.53◦ . Atoms are colored according to the potential energy, as indicated by the color bar. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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Fig. 5. The disregistry plot of the upper (lower) layer atoms in special boundaries: (a) and (b) = 26.53◦ , (c) and (d) = 31.59◦ , (e) and (f) = 38.94◦ , (g) and (h) = 50.48◦ , (i) and (j) = 70.53◦ . The length and color of arrows indicate the magnitude of the displacements, as indicated by the color bar. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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of special boundaries are relatively stable. And they reach stable states with little atomic displacements during relaxation (Fig. 5). As a result, special boundaries tend to have relatively low GBEs with an exception for ˙19 boundary, which is explained below. Boundaries in the vicinity of special boundaries have similar structures as the corresponding special boundary in local regions which improves their matching and reduce their GBEs. Fig. 4d–f are atomistic structures of the ˙3 TGB and boundaries in its vicinity. Combined with Fig. 1, we find that the atomistic structure gets close to that of ˙3 TGB gradually and the GBE decreases correspondingly as the approach of misorientation. The ordinary HAGBs are not influenced by special boundaries. They consist of periodic patterns and their GBEs near a stable value. Fig. 5 shows the displacements of upper and lower layer atoms of special boundaries during relaxation. As shown in Fig. 5, periodic rows or columns consisting of atoms which move towards each other are observed. And the positions of these rows or columns in upper and lower layers are interleaved with each other. According to these observations, we can make some conclusions. Before relaxation, some atoms in the upper and lower layers coincide with each other or close to each other due to the good symmetry of special boundaries. The strong repulsive force makes them move away from each other which reduce their GBEs. However, in ˙19 TGB, some atoms in the same layer come too close to each other after relaxation which can be seen in Fig. 5a and b where the red and yellow arrows point to almost same positions. The repulsion between these atoms in the same layer increases the GBE of ˙19 TGB. In other special boundaries (Fig. 5c–j), atoms in the same layer do not come so close and produce strong repulsion. Therefore, their GBEs are relatively low. It’s worth noting that the GBE of ˙19 boundary after relaxation is much lower than that of ˙19 boundary before relaxation even through there is repulsion. 4. Conclusion In summary, we study the energy and structure of (1 1 0) TGB in W over the full range of twist angles through molecular statics simulation. A dependence of GBE vs. the twist angle is obtained and agrees well with the improved Read–Shockley equation in lowangle range. The TGBs can be divided into three types: low-angle grain boundaries (LAGBs), with twist angle ≤ 16.10◦ , contain an intact dislocation network consisting of 12 [1 1 1], 21 [1¯ 1¯ 1] and [0 0 1] screw dislocations, which coincides with observations in experiment. The TGBs with 17.23◦ ≤ ≤ 22.22◦ are intermediate-angle grain boundaries (IAGBs), where both atomistic structures and dislocations are disordered and dislocations do not form a regular network. The TGBs with ≥ 23.5◦ are high-angle grain boundaries (HAGBs), where no dislocation is observed. The HAGBs can be divided into three sub-types further: special boundaries with low ˙, boundaries in their vicinity with similar structures as the corresponding special boundary in local regions as well as ordinary HAGBs consisting of periodic patterns. Acknowledgement This work is supported by the National Magnetic Confinement Fusion Program through Grant No. 2013GB109002.
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References [1] X.M. Wei, J.M. Zhang, K.W. Xu, The energy and structure of (0 0 1) twist grain boundary in noble metals, Appl. Surf. Sci. 253 (2006) 854–858. [2] X.M. Wei, J.M. Zhang, K.W. Xu, The periodicity in translation of Ag (0 0 1) and (1 1 0) twist grain boundary, Appl. Surf. Sci. 253 (2007) 4307–4310. [3] X.M. Wei, J.M. Zhang, K.W. Xu, Energy calculation of (0 1 1) twist grain boundary in noble metals, Appl. Surf. Sci. 252 (2006) 7331–7336. [4] J.M. Zhang, X.M. Wei, H. Xin, Energy analysis for (1 1 1) twist grain boundary in noble metals, Appl. Surf. Sci. 243 (2005) 1–6. [5] N.A. Gjostein, F.N. Rhines, Absolute interfacial energies of [0 0 1] tilt and twist grain boundaries in copper, Acta Metall. 7 (1959) 319–330. [6] A. Seki, O. Hellman, S. Tanaka, Calculation of grain boundary energies and structures in copper [0 0 1] twist boundaries using the modified embedded atom method, Scripta Mater. 34 (1996) 1867–1870. [7] D. Wolf, Effect of interatomic potential on the calculated energy and structure of high-angle coincident site grain boundaries – II. (1 0 0) twist boundaries in Cu, Ag and Au, Acta Metall. 32 (1984) 735–748. [8] A. Otsuki, Energies of (0 0 1) twist grain boundaries in silicon, Acta Mater. 49 (2001) 1737–1745. [9] M.R. Fitzsimmons, S.L. Sass, Quantitative X-ray diffraction study of the atomic structure of the ˙ = 5( = 36.9◦ ) [0 0 1] twist boundary in gold, Acta Metall. 36 (1988) 3103–3122. [10] P.D. Bristowe, S.L. Sass, The atomic structure of a large angle [0 0 1] twist boundary in gold determined by a joint computer modelling and X-ray diffraction study, Acta Metall. 28 (1980) 575–588. [11] M.R. Fitzsimmons, S.L. Sass, The atomic structure of the ˙ = 13( = 22.6◦ ) [0 0 1] twist boundary in gold determined using quantitative X-ray diffraction techniques, Acta Metall. 37 (1989) 1009–1022. [12] D. Wolf, Effect of interatomic potential on the calculated energy and structure of high-angle coincident site grain boundaries – I. (1 0 0) twist boundaries in aluminum, Acta Metall. 32 (1984) 245–258. [13] S. Dai, Y. Xiang, D.J. Srolovitz, Atomistic, generalized Peierls–Nabarro and analytical models for (1 1 1) twist boundaries in Al, Cu and Ni for all twist angles, Acta Mater. 69 (2014) 162–174. [14] S. Dai, Y. Xiang, D.J. Srolovitz, Structure and energy of (1 1 1) low-angle twist boundaries in Al, Cu and Ni, Acta Mater. 61 (2013) 1327–1337. [15] J.B. Yang, Y. Nagai, M. Hasegawa, Yu.N. Osetsky, Atomic scale modeling of {1 1 0} twist grain boundaries in ˛-iron: structure and energy properties, Philos. Mag. 90 (2010) 991–1000. [16] J.B. Yang, Y.N. Osetsky, R.E. Stoller, Y. Nagai, M. Hasegawa, The effect of twist angle on anisotropic mobility of {1 1 0} hexagonal dislocation networks in ˛-iron, Scripta Mater. 66 (2012) 761–764. [17] V.V. Bulatov, W. Cai, Nodal effects in dislocation mobility, Phys. Rev. Lett. 89 (2002) 115501. [18] S.J. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys. 117 (1995) 1–19. [19] N. Juslin, B.D. Wirth, Interatomic potentials for simulation of He bubble formation in W, J. Nucl. Mater. 432 (2013) 61–66. [20] A.V. Barashev, H. Xu, R.E. Stoller, The behavior of small helium clusters near free surfaces in tungsten, J. Nucl. Mater. 454 (2014) 421–426. [21] L. Hu, K.D. Hammond, B.D. Wirth, D. Maroudas, Dynamics of small mobile helium clusters near tungsten surfaces, Surf. Sci. 626 (2014) L21–L25. [22] F. Sefta, N. Juslin, K.D. Hammond, B.D. Wirth, Molecular dynamics simulations on the effect of sub-surface helium bubbles on the sputtering yield of tungsten, J. Nucl. Mater. 438 (2013) S493–S496. [23] J. Li, AtomEye: an efficient atomistic configuration viewer, Modell. Simul. Mater. Sci. Eng. 11 (2003) 173–177. [24] A. Stukowski, K. Albe, Extracting dislocations and non-dislocation crystal defects from atomistic simulation data, Modell. Simul. Mater. Sci. Eng. 18 (2010) 085001. [25] W.T. Read, W. Shockley, Dislocation models of crystal grain boundaries, Phys. Rev. 78 (1950) 275–289. [26] V. Vitek, On the difference between the misorientation dependences of the energies of tilt and twist boundaries, Scripta Metall. 21 (1987) 711–714. [27] S.M. Ohr, D.N. Beshers, Crystallography of dislocation networks in annealed iron, Philos. Mag. 8 (1963) 1343–1360. [28] A.J. Garratt-reed, G. Taylor, Optical and electron microscopy of niobium crystals deformed below room temperature, Philos. Mag. A 39 (1979) 597–646. [29] H.F. Ryan, J. Suiter, Observations of dislocations in high angle boundaries in tungsten, Acta Metall. 14 (1966) 847–851. [30] W. Bollmann, Crystal Defects and Crystalline Interfaces, Springer, Berlin, 1970. [31] A. Brokma, R.W. Balluffi, Coincidence lattice model for the structure and energy of grain boundaries, Acta Metall. 29 (1981) 1703–1719.