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(122) symmetric tilt grain boundary in Cu
Materials Chemistry and Physics 74 (2002) 313–319
HRTEM investigation of the multiplicity of Σ = 9 [0 1¯ 1]/(1 2 2) symmetric tilt grain boundary in Cu J.R. Hu a,∗ , S.C. Chang a , F.R. Chen b , J.J. Kai b a
Department of Material Science and Engineering, National Tsing Hua University, Hsinchu 300, Taiwan, ROC Department of Engineering and System Science, National Tsing Hua University, Hsinchu 300, Taiwan, ROC
b
Received 2 May 2001; received in revised form 7 July 2001; accepted 10 July 2001
Abstract Σ = 9 [0 1¯ 1]/(1 2 2) symmetric tilt grain boundary in Cu is investigated using high resolution transmission electron microscopy (HRTEM). Modified mirror and glide-mirror symmetric structural units (SUs) are found to coexist in the grain boundary. Partial dislocations with Burgers vectors close to half a displacement shift complete (DSC) lattice are introduced to connect the glide-mirror symmetric structural unit (SU) with the modified mirror symmetric one in the grain boundary. The Burgers vectors are analyzed to be close to a/18 [1 2 2]L // a/18 [1 2 2]R and a/36 [4¯ 1 1]L // a/36 [4¯ 1 1]R . The volume expansion per unit area of the grain boundary, δV/A, in the glide-mirror symmetric region is determined as −0.00238 nm which is a little smaller than that in rigid-body model, while theδV/A in the modified mirror symmetric region is computed as 0.03221 nm which is larger than that in the same model. The two values of δV/A are corresponding to compressive and tensile stress field, respectively. One possible mirror and glide-mirror symmetric structural model is proposed according to atomic relaxations. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Transmission electron microscopy (TEM); Bicrystal; Symmetric tilt; Grain boundary; Copper
1. Introduction Atomic structures of various Σ (inverse density of coincidence-sites) symmetric tilt grain boundaries (STGBs) in fcc and bcc metals have been investigated extensively using high resolution transmission electron microscopy (HRTEM) [1–12]. Mills [6] has found steps corresponding to the locations of the grain boundary dislocations (GBDs) in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Al. These GBDs associated with steps in the boundary are displacement shift complete (DSC) lattice dislocations referred to as intrinsic in nature [1]. The perfect DSC dislocation which could dissociate into two partial-DSC dislocations between mirror symmetric SU and glide-mirror symmetric SU in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Al [6] were also found. There may be a little deviation of misorientation from the exact Σ = 9 [0 1¯ 1]/(1 2 2) STGB. The Burgers vector of partial-DSC in Al [6] was not determined in the previous work. The coexistence of two boundary SUs was also observed in the case of Σ = 7 (0 1 1¯ 2) grain boundary in Al2 O3 [13]. In that case, possible Burgers vector of a partial-DSC dislocation was analyzed using geomet∗ Corresponding author. E-mail address: [email protected] (J.R. Hu).
rical model of coincidence-site lattice (CSL) dichromatic pattern. It seems that the structural multiplicity is always associated with a grain boundary dislocation. The energy of the mirror and glide-mirror symmetric SUs in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB have been calculated as 310 and 290 mJ m−2 , respectively, for Al [6]. Nevertheless, no multiplicity was proposed in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu from embedded atomic method (EAM) computer simulation [14]. Either, multiplicity was not found experimentally in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu in our previous report [15]. The energy of glide-mirror symmetric structure has been reported as 756 mJ m−2 for Cu [4,16]. Possible postulation is that the mirror symmetric SU is relatively energetically unstable in Cu due to higher repulsive energy arising from two closer atoms in the mirror symmetric SU [15]. The energy difference between mirror and glide-mirror symmetric SU for Cu must be larger than that for Al owing to the lower stacking fault energy (SFE) and higher bulk modulus [17] of Cu. The detail explanation was given in our previous publication [15]. This explanation is consistent with the higher stacking fault nature of Cu which is usually associated with larger shear modulus [15]. However, in this paper, we report coexistence of glidemirror and modified mirror symmetric SUs. The modified mirror symmetric unit is basically the same as the mirror
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one, except that one atom being removed from the unit in order to reduce the repulsive energy. Discussion of the relationships between multiplicity, GB energy, atomic relaxation and volume expansion will be given. Comparison with Al will also be undertaken.
2. Experiment procedures The single crystal of Cu is grown using vertical Bridgman technique. An electrolytic Cu bulk was melted at 1100 ◦ C in a graphite shell [18] and then was maintained at 1088 ◦ C with the furnace elevated vertically at a speed of 4.76 × 10−3 mm s−1 . The Cu single crystal was mounted on a goniometer that could be adopted to both Laue camera and polishing machine. After the tilt axis and boundary planes were polished, a Σ = 9 [0 1¯ 1]/(1 2 2) tilt bicrystal of Cu was prepared using diffusion bonding at a pressure of 16 kPa and a temperature 950 ◦ C for 24 h. The bicrystal was sliced with the normal of the slices parallel to the tilt axis [0 1¯ 1] direction. The slices were ground down to a thickness of about 80 m and then were punch to 3 mm disks. The disks were jet-polished in a 42.5 wt.% H3 PO4 ethanol solution at 25 V and 0 ◦ C. Finally, the polished disks were thinned with ion-mill at 5 kV and 1 mA. High resolution images were recorded using a JOEL 4000EX transmission electron microscope (TEM) which was operated at voltage 400 kV with a defocus value near Scherzer defocus −48 nm. Fourier Filter image processing [19] and maximum entropy method (MEM) [20] were employed to reduce the noise of HRTEM images. The projected positions of the atoms were determined by finding the peak intensities for every bright spot in the HRTEM image.
3. Results and discussion A HRTEM image of Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu is shown in Fig. 1. MEM is used to reduce the noise in the image [20]. The projected positions of the atoms in the HRTEM image are determined using a routine developed by Kilaas [21] which runs in DigitalMicrograph 2.5. We have found out that the MEM and Fourier filtered image give almost the same intensity maxima. Every position of atomic peaks is marked “cross” in Fig. 1. The simulated images (defocus −48 nm and thickness 10 nm) of glide-mirror and modified mirror symmetric structures are illustrated at the top and the bottom of the experimental image, respectively. The modified mirror and glide-mirror symmetric SUs are outlined with white lines. The (1 2 2) planes of both grains are labeled “A1A2”, “B1B2” and “C1C2”, respectively. There are two glide-mirror symmetric SUs and four modified mirror symmetric SUs in Fig. 1. The glide-mirror SU is similar to those observed experimentally in Al [6] and in Cu [15], however, the mirror symmetric SU is slightly different from that observed in Al. We called the symmetric SU in
Fig. 1. MEM image of HRTEM in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu. The mirror and glide-mirror symmetric SUs are outlined with white lines. The boundary planes (1 2 2) are labeled “A1A2”, “B1B2” and “C1C2”, respectively. The positions of the peak intensities are also marked with black crosses and superimposed on the image. The simulated images (defocus −48 nm and thickness 10 nm) of glide-mirror and mirror symmetric structures are illustrated at the top and the bottom, respectively.
Cu modified mirror symmetric SU. Fig. 2(a)–(d) show four schematic SUs of the Σ = 9 [0 1¯ 1]/(1 2 2) STGB. Fig. 2(a) and (b) are mirror and glide-mirror symmetric SUs, observed in Al, respectively, while Fig. 2(c) and (d) represent the same projected SUs, but with a rigid-body translation (RBT) of a/4 [0 1¯ 1] of the right-hand grain along the tilt axis direction. From [0 1¯ 1] projection, it is impossible to distinguish the difference in Fig. 2(a) and (c), as well as Fig. 2(b) and (d). It is worth pointing out two major differences in the modified mirror symmetric SU comparing with those in Fig. 2(a) and (c). The atoms marked #2 in Fig. 2(a) and (c) are missing in the modified mirror symmetric SU in Fig. 1.
J.R. Hu et al. / Materials Chemistry and Physics 74 (2002) 313–319
Fig. 2. Atomic structures of: (a) the Σ = 9 [0 1¯ 1]/(1 2 2) symmetric tilt grain boundary in high SFE Al proposed by Mills [6]; (b) the same boundary in low SFE fcc metals simulated by Rittner and Seidman [14]; (c and d) the same boundaries as (a and b) coupling with a rigid-body shift of a/4 [0 1¯ 1] of the right-hand grain. The model in (c) is proposed by the authors and the model in (d) by Rittner and Seidman [22]. The white circles represent the atoms locating on the paper page, while the black ones on the level of ±a/4 [0 1¯ 1].
Rittner and Seidman [14] have proposed that the mirror symmetric SU could only exist in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in high SFE fcc metals, i.e. Al, but not in low SFE fcc metals, i.e. Cu. The energy of SUs in Fig. 2(a) and (b) were calculated to be 310 and 290 mJ m−2 , respectively, and could coexist in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Al associated with dissociated a partial-DSC dislocation [6]. Wolf and Merkle have calculated the energy of glide-mirror symmetric SU as 756 mJ m−2 [4] for Cu, while the energies of the mirror symmetric SU in Cu has not been reported yet. We could expect that the mirror symmetric SU may associated with an energy higher than 756 mJ m−2 . The existence of modified mirror symmetric SU observed in Fig. 1 can be understood from the following discussion. The atomic configurations of Al and Cu are 3s2 3p1 and 3d10 4s1 , respectively. Daw and co-workers [23] has developed an approximation (well known as EAM) for the total energy of the form: Etotal =
1 Fi (ρh,i ) + Φij (Rij ), 2 i
i
i = j
j
where Φ ij (Rij ) is the core-to-core pair repulsion between atoms i and j separated by the distance Rij and is dependent on the effective charges of atoms i and j. The effective charge of Cu (ns + nd = 11) may be much larger than that of Al (ns + np = 3) so that the absolute differential value of dΦ ij (Rij )/dRij of Cu should be also larger than that of Al. It means that the repulsive energy between the atoms “2” and “×” in Fig. 2(a) and (c) in Cu may increase more dramatically than in Al. The higher bulk modulus of Cu (1.37 × 1011 N m−2 ) than of Al (0.722 × 1011 N m−2 ) [17,24] also corresponds with this postulation. Smaller effective charge and bulk modulus imply a flat bonding-energy
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curve near the equilibrium position, which suggests that Al may have more tolerance for two atoms being closer than their equilibrium distance to make the mirror symmetric SU present. According to these statements, the GB energy of the mirror symmetric structure in Cu may be much higher than 756 mJ m−2 so that the mirror symmetric structure becomes impossible to be present in equilibrium state. The repulsive energy could be reduced by removing (or partially removing) the atoms “2” in Fig. 2(a) and (c) to form a modified mirror symmetric SU which is found in our experiment (see Fig. 1). In fact, the SU in Fig. 2(a) and (c) will become similar to those in Fig. 2(b) and (d) if the atoms “×” are removed. The energy of the modified mirror symmetric structure may be close to 756 mJ m−2 . A dislocation is usually required to separate the co-existence of mirror and glide-mirror symmetric interfacial domains. Co-existence of two interfacial domains has been reported in Al [6] and Al2 O3 [13] experimentally. The g · b = 0 technique or a tedious simultaneous two beam condition coupling with computer simulation are usually applied to determine the Burgers vector experimentally [25]. Due to the geometry of bicrystal samples, application of the g · b = 0 technique or simultaneous two beam condition has a tilt limitation. Firstly, the Burgers vectors of grain boundary dislocations (a/18 [1 2 2]L // a/18 [1 2 2]R and a/36 [4¯ 1 1]L // a/36 [4 1 1]R for our case) are not simple lattice vectors. It is difficult to find proper g to satisfy g · b = 0 criterion. Secondly, we need common g’s in both grains to make g · b = 0 invisible criterion. For an edge-on boundary the condition is rare. The possible Burgers vector of dislocation is then analyzed from the CSL model in our paper. Fig. 3 shows a dichromatic pattern of a Σ = 9 [0 1¯ 1]/ (1 2 2) CSL lattice. The glide-mirror symmetric SU were outlined with the (1 1 2) plane which coincides with “A1A2” and “B1B2”. The plane “B1B2” locates at the coincident sites while the plane “A1A2” does not. Before the DSC dislocation was introduced into the boundary, “C1C2” plane is different from “D1D2” (1 2 2) plane. It should be noted that the symmetric (1 2 2) plane “D1D2” will be identical to the symmetric plane “C1C2” when the CSL “D1D2D3D4” shifts for a partial-DSC vector of a/36 [2¯ 5 5]L // a/12 [2 1 1]R , due to the nature of DSC having to preserve the dichromatic pattern. The perfect DSC vectors for cSTGB are DSC1 = a/9 [1 2 2]L // a/9 [1 2 2]R and DSC2 = a/18 [4¯ 1 1]L // a/18 [4 1 1]R , respectively. This partial-DSC vector is the combination of two mutually perpendicular partial-DSC vectors of the Σ = 9 [0 1¯ 1]/(1 2 2) STGB. b = a/18 [1 2 2]L + a/36 [4¯ 1 1]L = a/36 [2¯ 5 5]L , or b = a/18 [1 2 2]R + a/36 [4¯ 1 1]L = a/12 [2¯ 1 1]R By removing “+”from the right-hand crystal and “䊊” from the left-hand crystal in Fig. 3, we could obtain the structural model of co-existence of mirror and glide-mirror symmetric SUs. The structural model in the
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Fig. 3. The CSL lattice of the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu. “+” represents the atomic position of the left-hand grain and “䊊” represents that of the right-hand grain. The project atomic positions are the same as those in Fig. 1, without considering the relaxation near the GB.
Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu are given in Fig. 4(a) and (b). The right-hand grain in Fig. 4(b) has a rigid-body shift of a/4 [0 1 1] with respect to that in Fig. 4(a). Regardless of atomic relaxation, the GB structures in Fig. 4(a) and (b) are the same as those in Fig. 1 in terms of the projected atomic structures. The mirror and glide-mirror symmetric structures corresponding to those in Fig. 1 are outlined with black lines. Two partial-DSC dislocations “b1 ” and “b2 ” are given as a/18 [1 2 2]L and a/36 [4¯ 1 1]L , or a/18 [1 2 2]R and a/36 [4¯ 1 1]R , respectively. A step vector is usually associated with DSC dislocations [6] which correspond to the change of the position of the grain boundary plane. As shown in Fig. 3, the step vector “B2C1” is calculated as a/36 [3 4 3 1 3 1]L , or a/4 [6 1 1]R (5/2 DSC1 and 11/2 DSC2 ). Beware that the Burgers vector deduced from the CSL dichromatic pattern is only explanatory and qualitative. The real Burgers vector of dislocations may be close to the DSC vectors given here. The outlined mirror and glide-mirror symmetric SUs in Fig. 4 are composed of the rigid-body atomic positions, so that the distance between the planes “A1A2” and “B1B2” can be defined as two {2 4 4} lattice spacings which is equivalent to a vector of a/9 [1 2 2]. C1, C2, D1 and D2 in Fig. 3 are then regarded as on the same {2 4 4} plane and at coincident sites.
Fig. 4. (a) The DSC lattices of the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu. The GB structures are corresponding to those in Fig. 1. Two partial-DSC dislocations are labeled “b1 ” and “b2 ” which equal to (1/2) DSC1 = a/18 [1 2 2]L // a/18 [1 2 2]R and (1/2) DSC2 = a/36 [4¯ 1 1]L // a/36 [4¯ 1 1]R , respectively. (b) The same with (a) except for the rigid-body shift of a/4 [0 1¯ 1] of the right-hand grain.
J.R. Hu et al. / Materials Chemistry and Physics 74 (2002) 313–319
Fig. 5. The displacement map of the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu. The open circles A3, A4, B3, B4, C3, C4, C5, and C6 represent the rigid-body atoms, and the ends of the arrows indicate the positions of the observed atoms.
The relaxation near the boundary cores in [0 1¯ 1] projection after diffusion bonding can be studied by least-square fitting the positions of atomic peaks far away from the boundary in Fig. 1. It is worth mentioning that the relaxation here is not compared with the CSL state, but with reference of two perfect rigid crystal states. Fig. 5 shows the displacement map of the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu. The positions of A3, A4, B3, B4, C3, C4, C5, and C6 are the fixed points corresponds to unrelaxed crystal far from the boundary core. In order to highlight the SUs in the boundary core, the displacement vectors in the boundary core are also outlined. Fig. 6(a) and (b) are the plots of average displacement versus number of {2 4 4} planes away from the boundary cores of the glide-mirror and mirror symmetric structures, respectively. As we expected that the displacement decays to zero away from the boundary core where it is between the planes “A1A2” and “B1B2” for the glide-mirror, or on the plane “C1C2” for the mirror symmetric structure. The planes “A3A4” and “B3B4” in Fig. 5 represent the 19th {2 4 4} planes of the glide-mirror symmetric structure, while “C3C4” and “C5C6” depict the 18th {2 4 4} planes of the mirror symmetric structure that are close to the lattice planes in the bulk crystal. Fig. 6(a) shows two relatively large average relaxation on the first and fourth {2 4 4}
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planes in the left-hand grain. It is reasonable for the atoms on the first and fourth {2 4 4} planes in the left-hand grain to relax away from the boundary core because the distance between the atoms “A1” and “B5” is calculated as 0.24099 nm (2a/9 [1 2 2]) which is less than the equilibrium distance 0.25561 nm (see Fig. 4(b)). Therefore, the atoms “A5” should also relax away from the boundary core. Furthermore, the atoms “A6” and “A7” in Fig. 4(b) translate to “B1” coincidentally, and thus, possess relatively smaller relaxations. It also implies that the structural model in Fig. 4(b) may be the possible structure for the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu. The maximum relaxation in Fig. 6(b) appears on the second {2 4 4} plane in the right-hand grain. It is related to the fact that the atom “C7” in Fig. 4 is missing from the equivalent position in Fig. 1. The volume expansion per unit area of the GB, δV/A, can be obtained from Fig. 1. The distance between “A3A4” and “B3B4” planes are directly measured as 2.28699 nm by the proportional scale regardless the deviation of misorientation from the exact Σ = 9 [0 1¯ 1]/(1 2 2), while the distance between “A1A2” and “A3A4” planes are calculated as 1.08444 nm because there are 18 {2 4 4} lattice spacings between them. In the same way, the distance between “B1B2” and “B3B4” planes are also calculated as 1.08444 nm and the distance between “A1A2” and “B1B2” planes as 0.12049 nm, two {2 4 4} lattice spacings. Hence, the δV/A in the region of the glide-mirror symmetric structure is then calculated as −0.00238 nm. Equally, the distance between “C3C4” and “C5C6” planes are measured as 2.20109 nm, and then δV/A in the region of the mirror symmetric structure is computed as 0.03221 nm which would have been zero for ideal mirror symmetric structure. According to the results, one can conclude that the region near the glide-mirror symmetric structure is under a compressive stress field, while the region near the mirror symmetric structure under a tensile stress field. The fact that the distance between the planes “A3A4” and “B3B4” is less than 38 {2 4 4} lattice spacings is corresponding to a compressive stress field. For the same reason, the region of the modified mirror symmetric structure is yielded to a tensile stress field so that the distance between the planes “C3C4” and “C5C6” is larger than 36 {2 4 4} lattice spacings. As shown in Fig. 3, glide-mirror symmetric structure translates to mirror symmetric structure through a RBT vector a/36 [2¯ 5 5]L or a/12 [2 1 1]R , and forms two partial-DSC dislocations, b1 and b2 , of which the Burgers vectors are a/18 [1 2 2]L and a/36 [4¯ 1 1]L , or a/18 [1 2 2]R and a/36 [4¯ 1 1]R , respectively, as shown in Fig. 4(a) and (b). It is very clear now that the regions of the top-left lattices in Fig. 2(a) and (b) have the strongest compressive stress field, while the regions of the top-right and low-right lattices have relatively low compressive and tensile stress fields, respectively. Fig. 6(a) and (b) also show that the atoms in the left-hand grain have relatively higher relaxation displacements except for that on the second {2 4 4} plane of the modified mirror symmetric structure in the right-hand grain.
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Fig. 6. The plot of average displacements vs. the number of {2 4 4} plane away from the boundary cores in the regions of: (a) the glide-mirror symmetric structure and (b) the mirror symmetric structure. The average relaxation on the first {2 4 4} plane of the left-hand grain in (b) is regarded as zero because the atoms are missing here. The boundary core is on the zeroth {2 4 4} plane.
A large relaxation displacement may be corresponding to a strong stress field, either compressive or tensile. 4. Conclusions The modified mirror and glide-mirror symmetric SUs coexist in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB was reported in Cu. A partial-DSC dislocations are found to connect the glide-mirror symmetric SU with the modified mirror symmetric one in the GB. The Burgers vectors are analyzed to be close to a/36 [2¯ 5 5]L or a/12 [2 1 1]R , and the step vector is calculated as a/36 [3 4 3 1 3 1]L and a/4 [6 1 1]R . The volume expansion per unit area of the GB, δV/A, in the glide-mirror symmetric region is calculated as −0.00238 nm, while theδV/A in the modified mirror symmetric region is computed as 0.03221 nm which would have been zero for ideal mirror symmetric structure. The experimental δV/A which is smaller than that in the region of the glide-mirror symmetric structure in the rigid-body model exhibits a compressive stress field, while theδV/A
which is larger than that in the region of the modified mirror symmetric structure in the rigid-body model demonstrates a tensile stress field. A large relaxation displacement may be corresponding to a strong compressive or tensile stress field. One possible mirror and glide-mirror symmetric structural model with an a/4 [0 1¯ 1] translation relative to that simulated by Rittner et al. [8] is proposed, which is in conflict with the result of calculation of GB energy by Rittner and Seidman [22]. Acknowledgements Authors are grateful for the support of this research by the National Science Council, Republic of China, under Grant NSC 89–2216-E-007–027 and NSC 89–2216-E-007–035. References [1] A.P. Sutton, R.W. Balluffi, Acta Metall. 35 (1987) 2177. [2] W. Krakow, Acta Metall. Mater. 38 (1990) 1031.
J.R. Hu et al. / Materials Chemistry and Physics 74 (2002) 313–319 [3] J.M. Pénisson, U. Dahmen, M.J. Mills, Philos. Mag. Lett. 64 (1991) 277. [4] K.L. Merkle, D. Wolf, Philos. Mag. A 65 (1992) 513. [5] W. Krakow, Acta Metall. Mater. 40 (1992) 977. [6] M.J. Mills, Mater. Sci. Eng. A 166 (1993) 35. [7] D. Holfmann, F. Ernst, Ultramicroscopy 53 (1994) 205. [8] J.D. Rittner, D.N. Seidman, K.L. Merkle, Phys. Rev. B 53 (1996) R4241. [9] O.H. Duparc, S. Poulat, A. Larere, J. Thilbault, L. Priester, Philos. Mag. A 80 (2000) 853. [10] K. Morita, H. Nakashima, Mater. Sci. Eng. A234-236 (1997) 1053. [11] G.H. Campbell, J. Belak, J.A. Moriarty, Acta Mater. 47 (1999) 3977. [12] G.H. Campbell, J. Belak, J.A. Moriarty, Scripta Mater. 43 (2000) 659. [13] F.R. Chen, C.C. Chu, J.Y. Wang, L. Chang, Philos. Mag. A 72 (1995) 529. [14] J.D. Rittner, D.N. Seidman, Phys. Rev. B 54 (1996) 6999.
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[15] J.R. Hu, S.C. Chang, F.R. Chen, J.J. Kai, Scripta Mater. 45 (2001) 463. [16] D. Wolf, Acta Metall. 38 (1990) 781. [17] J.P. Hirth, J. Lothe, Theory of Dislocations, McGraw Hill, New York, 1968, p. 41, 762 and 764. [18] T.S. Sheu, S.C. Chang, Mater. Sci. Eng. A 147 (1991) 81. [19] D.B. Williams, C.B. Carter, Transmission Electron Microscopy, Plenum Press, New York, 1996, p. 523. [20] L.D. Marks, Ultramicroscopy 62 (1996) 43. [21] R. Kilaas, Proceedings of the 49th EMSA Meeting, San Francisco, CA, 1991, p. 528. [22] J.D. Rittner, D.N. Seidman, Acta Mater. 45 (1997) 3191. [23] S.M. Foiles, M.I. Baskes, M.S. Daw, Phys. Rev. B 33 (1986) 7983. [24] C. Kittel, Introduction to Solid State Physics, 7th Edition, Wiley, New York, 1996. [25] A.K. Head, P. Humble, L.M. Clarebrough, A.J. Morton, C.T. Forwood, Computed Electron Micrograph and Defect Identification, North-Holland, New York, 1973.
HRTEM investigation of the multiplicity of Σ = 9 [0 1¯ 1]/(1 2 2) symmetric tilt grain boundary in Cu J.R. Hu a,∗ , S.C. Chang a , F.R. Chen b , J.J. Kai b a
Department of Material Science and Engineering, National Tsing Hua University, Hsinchu 300, Taiwan, ROC Department of Engineering and System Science, National Tsing Hua University, Hsinchu 300, Taiwan, ROC
b
Received 2 May 2001; received in revised form 7 July 2001; accepted 10 July 2001
Abstract Σ = 9 [0 1¯ 1]/(1 2 2) symmetric tilt grain boundary in Cu is investigated using high resolution transmission electron microscopy (HRTEM). Modified mirror and glide-mirror symmetric structural units (SUs) are found to coexist in the grain boundary. Partial dislocations with Burgers vectors close to half a displacement shift complete (DSC) lattice are introduced to connect the glide-mirror symmetric structural unit (SU) with the modified mirror symmetric one in the grain boundary. The Burgers vectors are analyzed to be close to a/18 [1 2 2]L // a/18 [1 2 2]R and a/36 [4¯ 1 1]L // a/36 [4¯ 1 1]R . The volume expansion per unit area of the grain boundary, δV/A, in the glide-mirror symmetric region is determined as −0.00238 nm which is a little smaller than that in rigid-body model, while theδV/A in the modified mirror symmetric region is computed as 0.03221 nm which is larger than that in the same model. The two values of δV/A are corresponding to compressive and tensile stress field, respectively. One possible mirror and glide-mirror symmetric structural model is proposed according to atomic relaxations. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Transmission electron microscopy (TEM); Bicrystal; Symmetric tilt; Grain boundary; Copper
1. Introduction Atomic structures of various Σ (inverse density of coincidence-sites) symmetric tilt grain boundaries (STGBs) in fcc and bcc metals have been investigated extensively using high resolution transmission electron microscopy (HRTEM) [1–12]. Mills [6] has found steps corresponding to the locations of the grain boundary dislocations (GBDs) in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Al. These GBDs associated with steps in the boundary are displacement shift complete (DSC) lattice dislocations referred to as intrinsic in nature [1]. The perfect DSC dislocation which could dissociate into two partial-DSC dislocations between mirror symmetric SU and glide-mirror symmetric SU in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Al [6] were also found. There may be a little deviation of misorientation from the exact Σ = 9 [0 1¯ 1]/(1 2 2) STGB. The Burgers vector of partial-DSC in Al [6] was not determined in the previous work. The coexistence of two boundary SUs was also observed in the case of Σ = 7 (0 1 1¯ 2) grain boundary in Al2 O3 [13]. In that case, possible Burgers vector of a partial-DSC dislocation was analyzed using geomet∗ Corresponding author. E-mail address: [email protected] (J.R. Hu).
rical model of coincidence-site lattice (CSL) dichromatic pattern. It seems that the structural multiplicity is always associated with a grain boundary dislocation. The energy of the mirror and glide-mirror symmetric SUs in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB have been calculated as 310 and 290 mJ m−2 , respectively, for Al [6]. Nevertheless, no multiplicity was proposed in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu from embedded atomic method (EAM) computer simulation [14]. Either, multiplicity was not found experimentally in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu in our previous report [15]. The energy of glide-mirror symmetric structure has been reported as 756 mJ m−2 for Cu [4,16]. Possible postulation is that the mirror symmetric SU is relatively energetically unstable in Cu due to higher repulsive energy arising from two closer atoms in the mirror symmetric SU [15]. The energy difference between mirror and glide-mirror symmetric SU for Cu must be larger than that for Al owing to the lower stacking fault energy (SFE) and higher bulk modulus [17] of Cu. The detail explanation was given in our previous publication [15]. This explanation is consistent with the higher stacking fault nature of Cu which is usually associated with larger shear modulus [15]. However, in this paper, we report coexistence of glidemirror and modified mirror symmetric SUs. The modified mirror symmetric unit is basically the same as the mirror
0254-0584/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 2 5 4 - 0 5 8 4 ( 0 1 ) 0 0 4 8 4 - 9
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one, except that one atom being removed from the unit in order to reduce the repulsive energy. Discussion of the relationships between multiplicity, GB energy, atomic relaxation and volume expansion will be given. Comparison with Al will also be undertaken.
2. Experiment procedures The single crystal of Cu is grown using vertical Bridgman technique. An electrolytic Cu bulk was melted at 1100 ◦ C in a graphite shell [18] and then was maintained at 1088 ◦ C with the furnace elevated vertically at a speed of 4.76 × 10−3 mm s−1 . The Cu single crystal was mounted on a goniometer that could be adopted to both Laue camera and polishing machine. After the tilt axis and boundary planes were polished, a Σ = 9 [0 1¯ 1]/(1 2 2) tilt bicrystal of Cu was prepared using diffusion bonding at a pressure of 16 kPa and a temperature 950 ◦ C for 24 h. The bicrystal was sliced with the normal of the slices parallel to the tilt axis [0 1¯ 1] direction. The slices were ground down to a thickness of about 80 m and then were punch to 3 mm disks. The disks were jet-polished in a 42.5 wt.% H3 PO4 ethanol solution at 25 V and 0 ◦ C. Finally, the polished disks were thinned with ion-mill at 5 kV and 1 mA. High resolution images were recorded using a JOEL 4000EX transmission electron microscope (TEM) which was operated at voltage 400 kV with a defocus value near Scherzer defocus −48 nm. Fourier Filter image processing [19] and maximum entropy method (MEM) [20] were employed to reduce the noise of HRTEM images. The projected positions of the atoms were determined by finding the peak intensities for every bright spot in the HRTEM image.
3. Results and discussion A HRTEM image of Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu is shown in Fig. 1. MEM is used to reduce the noise in the image [20]. The projected positions of the atoms in the HRTEM image are determined using a routine developed by Kilaas [21] which runs in DigitalMicrograph 2.5. We have found out that the MEM and Fourier filtered image give almost the same intensity maxima. Every position of atomic peaks is marked “cross” in Fig. 1. The simulated images (defocus −48 nm and thickness 10 nm) of glide-mirror and modified mirror symmetric structures are illustrated at the top and the bottom of the experimental image, respectively. The modified mirror and glide-mirror symmetric SUs are outlined with white lines. The (1 2 2) planes of both grains are labeled “A1A2”, “B1B2” and “C1C2”, respectively. There are two glide-mirror symmetric SUs and four modified mirror symmetric SUs in Fig. 1. The glide-mirror SU is similar to those observed experimentally in Al [6] and in Cu [15], however, the mirror symmetric SU is slightly different from that observed in Al. We called the symmetric SU in
Fig. 1. MEM image of HRTEM in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu. The mirror and glide-mirror symmetric SUs are outlined with white lines. The boundary planes (1 2 2) are labeled “A1A2”, “B1B2” and “C1C2”, respectively. The positions of the peak intensities are also marked with black crosses and superimposed on the image. The simulated images (defocus −48 nm and thickness 10 nm) of glide-mirror and mirror symmetric structures are illustrated at the top and the bottom, respectively.
Cu modified mirror symmetric SU. Fig. 2(a)–(d) show four schematic SUs of the Σ = 9 [0 1¯ 1]/(1 2 2) STGB. Fig. 2(a) and (b) are mirror and glide-mirror symmetric SUs, observed in Al, respectively, while Fig. 2(c) and (d) represent the same projected SUs, but with a rigid-body translation (RBT) of a/4 [0 1¯ 1] of the right-hand grain along the tilt axis direction. From [0 1¯ 1] projection, it is impossible to distinguish the difference in Fig. 2(a) and (c), as well as Fig. 2(b) and (d). It is worth pointing out two major differences in the modified mirror symmetric SU comparing with those in Fig. 2(a) and (c). The atoms marked #2 in Fig. 2(a) and (c) are missing in the modified mirror symmetric SU in Fig. 1.
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Fig. 2. Atomic structures of: (a) the Σ = 9 [0 1¯ 1]/(1 2 2) symmetric tilt grain boundary in high SFE Al proposed by Mills [6]; (b) the same boundary in low SFE fcc metals simulated by Rittner and Seidman [14]; (c and d) the same boundaries as (a and b) coupling with a rigid-body shift of a/4 [0 1¯ 1] of the right-hand grain. The model in (c) is proposed by the authors and the model in (d) by Rittner and Seidman [22]. The white circles represent the atoms locating on the paper page, while the black ones on the level of ±a/4 [0 1¯ 1].
Rittner and Seidman [14] have proposed that the mirror symmetric SU could only exist in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in high SFE fcc metals, i.e. Al, but not in low SFE fcc metals, i.e. Cu. The energy of SUs in Fig. 2(a) and (b) were calculated to be 310 and 290 mJ m−2 , respectively, and could coexist in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Al associated with dissociated a partial-DSC dislocation [6]. Wolf and Merkle have calculated the energy of glide-mirror symmetric SU as 756 mJ m−2 [4] for Cu, while the energies of the mirror symmetric SU in Cu has not been reported yet. We could expect that the mirror symmetric SU may associated with an energy higher than 756 mJ m−2 . The existence of modified mirror symmetric SU observed in Fig. 1 can be understood from the following discussion. The atomic configurations of Al and Cu are 3s2 3p1 and 3d10 4s1 , respectively. Daw and co-workers [23] has developed an approximation (well known as EAM) for the total energy of the form: Etotal =
1 Fi (ρh,i ) + Φij (Rij ), 2 i
i
i = j
j
where Φ ij (Rij ) is the core-to-core pair repulsion between atoms i and j separated by the distance Rij and is dependent on the effective charges of atoms i and j. The effective charge of Cu (ns + nd = 11) may be much larger than that of Al (ns + np = 3) so that the absolute differential value of dΦ ij (Rij )/dRij of Cu should be also larger than that of Al. It means that the repulsive energy between the atoms “2” and “×” in Fig. 2(a) and (c) in Cu may increase more dramatically than in Al. The higher bulk modulus of Cu (1.37 × 1011 N m−2 ) than of Al (0.722 × 1011 N m−2 ) [17,24] also corresponds with this postulation. Smaller effective charge and bulk modulus imply a flat bonding-energy
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curve near the equilibrium position, which suggests that Al may have more tolerance for two atoms being closer than their equilibrium distance to make the mirror symmetric SU present. According to these statements, the GB energy of the mirror symmetric structure in Cu may be much higher than 756 mJ m−2 so that the mirror symmetric structure becomes impossible to be present in equilibrium state. The repulsive energy could be reduced by removing (or partially removing) the atoms “2” in Fig. 2(a) and (c) to form a modified mirror symmetric SU which is found in our experiment (see Fig. 1). In fact, the SU in Fig. 2(a) and (c) will become similar to those in Fig. 2(b) and (d) if the atoms “×” are removed. The energy of the modified mirror symmetric structure may be close to 756 mJ m−2 . A dislocation is usually required to separate the co-existence of mirror and glide-mirror symmetric interfacial domains. Co-existence of two interfacial domains has been reported in Al [6] and Al2 O3 [13] experimentally. The g · b = 0 technique or a tedious simultaneous two beam condition coupling with computer simulation are usually applied to determine the Burgers vector experimentally [25]. Due to the geometry of bicrystal samples, application of the g · b = 0 technique or simultaneous two beam condition has a tilt limitation. Firstly, the Burgers vectors of grain boundary dislocations (a/18 [1 2 2]L // a/18 [1 2 2]R and a/36 [4¯ 1 1]L // a/36 [4 1 1]R for our case) are not simple lattice vectors. It is difficult to find proper g to satisfy g · b = 0 criterion. Secondly, we need common g’s in both grains to make g · b = 0 invisible criterion. For an edge-on boundary the condition is rare. The possible Burgers vector of dislocation is then analyzed from the CSL model in our paper. Fig. 3 shows a dichromatic pattern of a Σ = 9 [0 1¯ 1]/ (1 2 2) CSL lattice. The glide-mirror symmetric SU were outlined with the (1 1 2) plane which coincides with “A1A2” and “B1B2”. The plane “B1B2” locates at the coincident sites while the plane “A1A2” does not. Before the DSC dislocation was introduced into the boundary, “C1C2” plane is different from “D1D2” (1 2 2) plane. It should be noted that the symmetric (1 2 2) plane “D1D2” will be identical to the symmetric plane “C1C2” when the CSL “D1D2D3D4” shifts for a partial-DSC vector of a/36 [2¯ 5 5]L // a/12 [2 1 1]R , due to the nature of DSC having to preserve the dichromatic pattern. The perfect DSC vectors for cSTGB are DSC1 = a/9 [1 2 2]L // a/9 [1 2 2]R and DSC2 = a/18 [4¯ 1 1]L // a/18 [4 1 1]R , respectively. This partial-DSC vector is the combination of two mutually perpendicular partial-DSC vectors of the Σ = 9 [0 1¯ 1]/(1 2 2) STGB. b = a/18 [1 2 2]L + a/36 [4¯ 1 1]L = a/36 [2¯ 5 5]L , or b = a/18 [1 2 2]R + a/36 [4¯ 1 1]L = a/12 [2¯ 1 1]R By removing “+”from the right-hand crystal and “䊊” from the left-hand crystal in Fig. 3, we could obtain the structural model of co-existence of mirror and glide-mirror symmetric SUs. The structural model in the
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Fig. 3. The CSL lattice of the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu. “+” represents the atomic position of the left-hand grain and “䊊” represents that of the right-hand grain. The project atomic positions are the same as those in Fig. 1, without considering the relaxation near the GB.
Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu are given in Fig. 4(a) and (b). The right-hand grain in Fig. 4(b) has a rigid-body shift of a/4 [0 1 1] with respect to that in Fig. 4(a). Regardless of atomic relaxation, the GB structures in Fig. 4(a) and (b) are the same as those in Fig. 1 in terms of the projected atomic structures. The mirror and glide-mirror symmetric structures corresponding to those in Fig. 1 are outlined with black lines. Two partial-DSC dislocations “b1 ” and “b2 ” are given as a/18 [1 2 2]L and a/36 [4¯ 1 1]L , or a/18 [1 2 2]R and a/36 [4¯ 1 1]R , respectively. A step vector is usually associated with DSC dislocations [6] which correspond to the change of the position of the grain boundary plane. As shown in Fig. 3, the step vector “B2C1” is calculated as a/36 [3 4 3 1 3 1]L , or a/4 [6 1 1]R (5/2 DSC1 and 11/2 DSC2 ). Beware that the Burgers vector deduced from the CSL dichromatic pattern is only explanatory and qualitative. The real Burgers vector of dislocations may be close to the DSC vectors given here. The outlined mirror and glide-mirror symmetric SUs in Fig. 4 are composed of the rigid-body atomic positions, so that the distance between the planes “A1A2” and “B1B2” can be defined as two {2 4 4} lattice spacings which is equivalent to a vector of a/9 [1 2 2]. C1, C2, D1 and D2 in Fig. 3 are then regarded as on the same {2 4 4} plane and at coincident sites.
Fig. 4. (a) The DSC lattices of the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu. The GB structures are corresponding to those in Fig. 1. Two partial-DSC dislocations are labeled “b1 ” and “b2 ” which equal to (1/2) DSC1 = a/18 [1 2 2]L // a/18 [1 2 2]R and (1/2) DSC2 = a/36 [4¯ 1 1]L // a/36 [4¯ 1 1]R , respectively. (b) The same with (a) except for the rigid-body shift of a/4 [0 1¯ 1] of the right-hand grain.
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Fig. 5. The displacement map of the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu. The open circles A3, A4, B3, B4, C3, C4, C5, and C6 represent the rigid-body atoms, and the ends of the arrows indicate the positions of the observed atoms.
The relaxation near the boundary cores in [0 1¯ 1] projection after diffusion bonding can be studied by least-square fitting the positions of atomic peaks far away from the boundary in Fig. 1. It is worth mentioning that the relaxation here is not compared with the CSL state, but with reference of two perfect rigid crystal states. Fig. 5 shows the displacement map of the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu. The positions of A3, A4, B3, B4, C3, C4, C5, and C6 are the fixed points corresponds to unrelaxed crystal far from the boundary core. In order to highlight the SUs in the boundary core, the displacement vectors in the boundary core are also outlined. Fig. 6(a) and (b) are the plots of average displacement versus number of {2 4 4} planes away from the boundary cores of the glide-mirror and mirror symmetric structures, respectively. As we expected that the displacement decays to zero away from the boundary core where it is between the planes “A1A2” and “B1B2” for the glide-mirror, or on the plane “C1C2” for the mirror symmetric structure. The planes “A3A4” and “B3B4” in Fig. 5 represent the 19th {2 4 4} planes of the glide-mirror symmetric structure, while “C3C4” and “C5C6” depict the 18th {2 4 4} planes of the mirror symmetric structure that are close to the lattice planes in the bulk crystal. Fig. 6(a) shows two relatively large average relaxation on the first and fourth {2 4 4}
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planes in the left-hand grain. It is reasonable for the atoms on the first and fourth {2 4 4} planes in the left-hand grain to relax away from the boundary core because the distance between the atoms “A1” and “B5” is calculated as 0.24099 nm (2a/9 [1 2 2]) which is less than the equilibrium distance 0.25561 nm (see Fig. 4(b)). Therefore, the atoms “A5” should also relax away from the boundary core. Furthermore, the atoms “A6” and “A7” in Fig. 4(b) translate to “B1” coincidentally, and thus, possess relatively smaller relaxations. It also implies that the structural model in Fig. 4(b) may be the possible structure for the Σ = 9 [0 1¯ 1]/(1 2 2) STGB in Cu. The maximum relaxation in Fig. 6(b) appears on the second {2 4 4} plane in the right-hand grain. It is related to the fact that the atom “C7” in Fig. 4 is missing from the equivalent position in Fig. 1. The volume expansion per unit area of the GB, δV/A, can be obtained from Fig. 1. The distance between “A3A4” and “B3B4” planes are directly measured as 2.28699 nm by the proportional scale regardless the deviation of misorientation from the exact Σ = 9 [0 1¯ 1]/(1 2 2), while the distance between “A1A2” and “A3A4” planes are calculated as 1.08444 nm because there are 18 {2 4 4} lattice spacings between them. In the same way, the distance between “B1B2” and “B3B4” planes are also calculated as 1.08444 nm and the distance between “A1A2” and “B1B2” planes as 0.12049 nm, two {2 4 4} lattice spacings. Hence, the δV/A in the region of the glide-mirror symmetric structure is then calculated as −0.00238 nm. Equally, the distance between “C3C4” and “C5C6” planes are measured as 2.20109 nm, and then δV/A in the region of the mirror symmetric structure is computed as 0.03221 nm which would have been zero for ideal mirror symmetric structure. According to the results, one can conclude that the region near the glide-mirror symmetric structure is under a compressive stress field, while the region near the mirror symmetric structure under a tensile stress field. The fact that the distance between the planes “A3A4” and “B3B4” is less than 38 {2 4 4} lattice spacings is corresponding to a compressive stress field. For the same reason, the region of the modified mirror symmetric structure is yielded to a tensile stress field so that the distance between the planes “C3C4” and “C5C6” is larger than 36 {2 4 4} lattice spacings. As shown in Fig. 3, glide-mirror symmetric structure translates to mirror symmetric structure through a RBT vector a/36 [2¯ 5 5]L or a/12 [2 1 1]R , and forms two partial-DSC dislocations, b1 and b2 , of which the Burgers vectors are a/18 [1 2 2]L and a/36 [4¯ 1 1]L , or a/18 [1 2 2]R and a/36 [4¯ 1 1]R , respectively, as shown in Fig. 4(a) and (b). It is very clear now that the regions of the top-left lattices in Fig. 2(a) and (b) have the strongest compressive stress field, while the regions of the top-right and low-right lattices have relatively low compressive and tensile stress fields, respectively. Fig. 6(a) and (b) also show that the atoms in the left-hand grain have relatively higher relaxation displacements except for that on the second {2 4 4} plane of the modified mirror symmetric structure in the right-hand grain.
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Fig. 6. The plot of average displacements vs. the number of {2 4 4} plane away from the boundary cores in the regions of: (a) the glide-mirror symmetric structure and (b) the mirror symmetric structure. The average relaxation on the first {2 4 4} plane of the left-hand grain in (b) is regarded as zero because the atoms are missing here. The boundary core is on the zeroth {2 4 4} plane.
A large relaxation displacement may be corresponding to a strong stress field, either compressive or tensile. 4. Conclusions The modified mirror and glide-mirror symmetric SUs coexist in the Σ = 9 [0 1¯ 1]/(1 2 2) STGB was reported in Cu. A partial-DSC dislocations are found to connect the glide-mirror symmetric SU with the modified mirror symmetric one in the GB. The Burgers vectors are analyzed to be close to a/36 [2¯ 5 5]L or a/12 [2 1 1]R , and the step vector is calculated as a/36 [3 4 3 1 3 1]L and a/4 [6 1 1]R . The volume expansion per unit area of the GB, δV/A, in the glide-mirror symmetric region is calculated as −0.00238 nm, while theδV/A in the modified mirror symmetric region is computed as 0.03221 nm which would have been zero for ideal mirror symmetric structure. The experimental δV/A which is smaller than that in the region of the glide-mirror symmetric structure in the rigid-body model exhibits a compressive stress field, while theδV/A
which is larger than that in the region of the modified mirror symmetric structure in the rigid-body model demonstrates a tensile stress field. A large relaxation displacement may be corresponding to a strong compressive or tensile stress field. One possible mirror and glide-mirror symmetric structural model with an a/4 [0 1¯ 1] translation relative to that simulated by Rittner et al. [8] is proposed, which is in conflict with the result of calculation of GB energy by Rittner and Seidman [22]. Acknowledgements Authors are grateful for the support of this research by the National Science Council, Republic of China, under Grant NSC 89–2216-E-007–027 and NSC 89–2216-E-007–035. References [1] A.P. Sutton, R.W. Balluffi, Acta Metall. 35 (1987) 2177. [2] W. Krakow, Acta Metall. Mater. 38 (1990) 1031.
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