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Structure and climb of 13 111 twin dislocations in aluminum
Materials Science and Engineering A319– 321 (2001) 102– 106 www.elsevier.com/locate/msea
Structure and climb of 13 111 twin dislocations in aluminum S.M. Foiles a,*, D.L. Medlin b a
Sandia National Laboratories, Mail Stop 1411, P.O. Box 5800, Albuquerque, NM 87185, USA Sandia National Laboratories, Mail Stop 9161, P.O. Box 0969. Li6ermore, CA 94551, USA
b
Abstract The structure and motion via climb of 13 111 twin dislocations along a coherent twin boundary in aluminum are examined via a combination of high resolution transmission electron microscopy (HRTEM) and computer simulation. Climb of these defects increases the thickness of the twin and is a possible alternative to the more familiar glide mechanism. Detailed analysis of the observed defect structure and comparison with atomistic modeling shows that the 13 111 twin dislocations exist in a dissociated configuration. This structure can be interpreted as the relaxation of the 13 111 dislocation into two dislocations, a Shockley partial and a stair-rod dislocation, which are separated by a short segment of stacking fault. The atomistic mechanism of the climb of the dislocations is analyzed. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Structure; Twin dislocations; Aluminum
1. Introduction The formation of a twin boundary step by a gliding twin dislocation is well known (e.g. [1,2]). Perhaps less familiar is the formation of a FCC twin step by a dislocation of Burgers vector b = 13 111. This Burger’s vector is a member of the displacement shift complete (DSC) lattice [3], which represents the set of all translation vectors between the two perfect crystals. A schematic of this dislocation is shown in Fig. 1. This schematic illustrates that this dislocation corresponds to the termination of a single {111} plane of atoms. On the left side of the dislocation in Fig. 1, the {111} stacking sequence about the twin plane (indicated in bold face) is: CAB6 ACB. On the right side of the dislocation, the ‘A’ plane is removed giving a stacking sequence of CABC6 B, and thereby moving the twin boundary up by one (111) plane, producing a monolayer step. Direct, atomic scale evidence for such 1 3 111 dislocations at FCC twin boundaries has been obtained in high-resolution transmission electron microscopy (HRTEM) studies of aluminum [4–7] and gold [8]. * Corresponding author. Tel.: + 1-505-844-7064; fax: +1-505-8449781. E-mail address: [email protected] (S.M. Foiles).
Because the Burgers vector of the 13 111 dislocation is orthogonal to the twin plane, motion of this defect along the interface can occur only by climb. This is in contrast to the 16 [2( 11] dislocation, which moves by glide in the twin plane. We have recently reported in situ observations of such climb in an aluminum twin boundary under electron irradiation [5–7]. The structure of this boundary has been compared to atomistic simulation [7]. In the present paper, we review the detailed structure of the 13 111 dislocation and examine the energetics of vacancy formation in the boundary and the energetics associated with the climb process. We show that this defect relaxes significantly from the simple model depicted in Fig. 1, and that this relaxation can be understood in terms of a dislocation dissociation reaction. Further, the probable mechanism for the climb of the dislocation is examined. It is argued that the climb proceeds via a double kink formation process. 2. Observation of climbing 13 111 dislocations at a coherent twin interface The experimental observations presented here were made on a S= 3 (21( 1( )[01( 1] aluminum bicrystal, grown using a horizontal zone melting technique as described in Ref. [7]. This bicrystal was provided by C. Goux and
0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 1 ) 0 1 1 1 2 - 1
S.M. Foiles, D.L. Medlin / Materials Science and Engineering A319–321 (2001) 102–106
Fig. 1. Schematics showing the step in twin boundary produced by DSC lattice dislocation of Burgers vectors 13 [1( 1( 1( ]. Vertical lines of the same type are spaced at intervals of 16 [2( 11]. Horizontal lines are spaced at intervals of 13 [111]. Dashed lines are set into the plane of the figure by 14 [011( ].
M. Biscondi of the Ecole National Superieure des Mines, St. Etiennes, France. HRTEM observations of the boundary structure were conducted in a JEOL 4000EX electron microscope operated at 400 keV. As we have reported previously [5,7], a region of the S= 3 (21( 1( ) boundary roughened by the emission of narrow coherent twin lamellae during electron microscopic examination. Analysis of a sequence of micrographs taken during the growth of the lamella showed an array of steps associated with 13 111 dislocation on the top (111) facet that moved along the (111) facet during the growth of the twin. Emission and subsequent motion of the 13 111 dislocations along the coherent twin interface occurs by climb. The 400 keV electrons used for the observations are well above the 170 keV knock-on displacement threshold for Frenkel pair production in aluminum [9]. In general, such motion would also be conceivable under thermal processing especially at elevated temperatures. The experimental observations show that the structures relax significantly from the idealized model shown in Fig. 1. The {111} fringes near the step are offset such that the extra ‘half-fringe’ appears to be centered three (111) fringes above the twin plane. In contrast, the extra half-plane associated with the dislocation in the unrelaxed DSC lattice model is positioned immediately above the twin plane. The precise degree of this offset varies with offsets observed in the range of 2– 5 planes.
3. Interface modeling and comparison with HRTEM observations To better understand the interfacial dislocation relaxation, we have modeled the atomistic structure of a 1 3 111 twin dislocation using the embedded atom method [10,11]. The geometry of the calculations is described in detail elsewhere [7]. Fig. 2 shows a section of the relaxed structure in the vicinity of a 13 [1( 1( 1( ]
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Fig. 2. Relaxed structure for 13 [1( 1( 1( ] interfacial dislocation at twin interface in Aluminum. The dashed line indicates the apparent position of the extra {111} plane and the lines labeled CT denote the position of the twin boundaries.
dislocation illustrating the resulting step in the coherent twin boundary (marked CT). The calculation predicts a relaxation from the compact structure of the DSC model: specifically, the {111} planes in the vicinity of the twin step are offset such that the extra {111} plane appears centered three planes above the coherent twin interface. The center of this distortion is marked on the calculated structure by a horizontal dashed line. This offset is consistent with the average observed experimentally. To compare the theoretical structure with the experimentally observed step configurations, we conducted HRTEM image simulations using the EMS package of electron microscopy simulation software [12]. Fig. 3a shows a simulated HRTEM image calculated from the theoretically derived atomic positions. Imaging parameters are summarized in the figure caption. An experimental image of a twin step and interfacial dislocation is shown in Fig. 3b overlaid with the positions of the intensity maximums from the theoretical image. Overall, there is a good, qualitative correspondence between the experimental and calculated images. An important feature to note in the calculated structure is the segment of stacking fault that extends from the twin step into the upper crystal along a (11( 1( ) plane. This short fault is marked on Fig. 2 by a black and white line and corresponds to a jog in the horizontal {111} planes. That interfaces may dissociate by the emission of stacking faults has been discussed recently [13], and a large number examples of such interfacial decomposition at interfaces in low stacking fault energy metals have now been reported [14–17]. Of particular relevance to the situation here are recent observations and calculations of 13 111 dislocations at tilted Ag/ Mn3O4 and Ag/ZnO interfaces, which were found to dissociate producing stacking faults extending from the interface into the metal [18,19].
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Fig. 3. Comparison of calculated structure for twin defect with experimental HRTEM observation. (a) Simulated HRTEM image for calculated defect structure shown in Fig. 5. Image simulation parameters: electron energy: 400 keV; spherical aberration coefficient: 1 mm; focus spread: 10 nm; semi-convergence angle: 1.0 mrad; specimen thickness: 5.7 nm; and defocus: 70 nm. (b) Experimental HRTEM image of twin defect shown in Fig. 3. Intensity peak positions from the simulated image shown in Fig. 6(a) are overlaid on the image.
Although aluminum has a high stacking fault energy a related dissociation (though to a more limited extent) has occurred here. A Frank dislocation can dissociate into a Shockley partial dislocation and stair-rod partial dislocation separated by a stacking fault (e.g. Ref. [20]). This dislocation reaction plays a key role in the formation of stacking fault tetrahedra in FCC metals of low stacking fault energy (SFE). As illustrated in Fig. 4, the perfect 13 [1( 1( 1( ] DSC dislocation (dD in Thompson’s notation) may also dissociate by emitting from the interface a Shockley partial dislocation (b= 16 [2( 1( 1( ] =bD) onto the (11( 1( ) plane. This produces a stacking fault on the (11( 1( ) plane and leaves behind a stair-rod dislocation (b = 16 [01( 1( ]= db) at the twin step. Referring to the calculated structure shown in Fig. 2, the core of the stair-rod dislocation (db) corresponds to the triangle of identically shaded atoms at the twin step and the stacking fault is indicated by the inclined solid line. The dissociated structure of the dislocation raises questions concerning the details of the climb process. In particular, the continuum picture suggests that the dissociated configuration must constrict in order to absorb vacancies and thereby climb. In order to address this issue, the energy and structure of various vacancy configurations has been examined. Fig. 5 presents a view of the formation energy of single vacancies in the vicinity of the dislocation core. The shading reflects local value of the vacancy formation energy. For the bulk crystal, the computed value of Efv is 0.69 eV. This compares with the experimental value of 0.67 eV [21]. The vacancy formation energy near the dislocation ranges from a low of 0.53 eV to a high of 0.91 eV. The low formation energies occur at the cores of both the stair-rod and Shockley par-
tials. Thus a single vacancy would tend to occupy either of those core positions with comparable frequency. The situation is substantially different when multiple vacancies move to the dislocation core. The formation energy for adding additional vacancies to vacancy clusters is plotted in Fig. 6a for clusters up to six vacancies. The qualitative behavior is quite different for the two cores. For the case of a vacancy cluster at the Shockley partial core, the formation energy saturates at a value of 0.26 eV. For the stairrod core, the formation energy for additional vacancies actually approaches zero. Thus it is energetically favorable for multiple vacancies to bind at the stairrod core. In order to understand the behavior of the formation energy of vacancy clusters at the stair rod core, consider the structure of the six vacancy cluster shown in Fig. 6b. The important point is that the local structure in the central region of this cluster is equivalent to that found for the straight dislocation. Thus the energy associated with the vacancy cluster corresponds to the energy to form the kinks at the
Fig. 4. Schematic showing geometry for dissociation of 13 [1( 1( 1( ] interfacial dislocation (dD) into a stair-rod dislocation (b = 16 [01( 1( ] = db), which remains at interface, and a Shockley partial (b = 16 [2( 1( 1( ] = bD), which leaves a stacking fault on the (11( 1( ) plane.
S.M. Foiles, D.L. Medlin / Materials Science and Engineering A319–321 (2001) 102–106
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Fig. 5. Calculated values of the vacancy formation energies in the vicinity of the dislocation. The color represents the value of the vacancy formation energy with the lightest shade corresponding to 0.53 eV and the darkest shade to 0.91 eV.
edge of the cluster plus a small interaction energy between the kinks. Thus to add more vacancies to the cluster requires very little energy since this simply changes the separation of the kinks. Thus the climb of the dislocation is accomplished by a double kink mechanism. The main energy barrier is the initial formation of the double kink. The calculations also indicate that there is no energy barrier for the separation of the kinks as might be expected based on the dissociated nature of the core. Instead, the atomic structure relaxes without any barrier upon the addition of a vacancy.
4. Summary and concluding comments The presence of a 13 111 dislocation at a twin boundary produces a step that can, and does, move by climb. This is an important observation in that it provides a mechanism whereby twin growth can occur by non-conservative, thermally activated processes, rather than the conservative glide processes conventionally envisioned for twinning. The climb of the dislocation is seen to occur by a double kink type of mechanism. The formation of this double kink and its expansion does not appear to require an energy barrier. Such a barrier might be expected on continuum grounds if the dissociated dislocation needed to constrict in order to climb.
Acknowledgements This research is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under contract number DE-AC04-94-AL85000.
Fig. 6. (a) The vacancy formation energy in eV for the creation of the last vacancy of a cluster of size N. The triangles are for clusters at the Shockley core and the squares are at the stair-rod core. (b) The structure of a six vacancy cluster at the core of the stair-rod dislocation viewed along {111} directions. The shading represents the position normal the page. Note that the structure at the vacancy cluster is the same as for the straight boundary except at the two ends of the cluster.
References [1] F.C. Frank, J.H. van der Merwe, Proc. R. Soc. Lond. A198 (1949) 205 – 216. [2] W.T. Read, Dislocations in Crystals, McGraw-Hill, New York, 1953, p. 109. [3] W. Bollmann, Crystal Defects and Crystalline Interfaces, Springer-Verlag, New York, 1970. [4] M. Shamzuzzoha, P.A. Deymier, D.J. Smith, Philos. Mag. A64 (1991) 245 – 253. [5] D.L. Medlin, C.B. Carter, J.E. Angelo, M.J. Mills, Philos. Mag. A75 (1997) 733 – 747. [6] D.L. Medlin, S.M. Foiles, C.B. Carter, in: D.J. Smith (Ed.), Atomic Resolution Microscopy of Surfaces and Interfaces, MRS Symp. Proc, vol. 466, Materials Research Society, Pittsburgh, PA, 1997, pp. 125 – 130. [7] D.L. Medlin, in: S. Ankem, C.S. Pande (Eds.), Advances in Twinning, The Minerals, Metals and Materials Society, Warrendale, PA, 1999. [8] K.L. Merkle, J. Phys. Chem. Solids 55 (1994) 991 –1005. [9] A. Wolfenden, Radiat. Eff. 14 (1972) 225.
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[10] M.S. Daw, S.M. Foiles, M.I. Baskes, Mater. Sci. Rep. 9 (1993) 251. [11] F. Ercolessi, J.B. Adams, Europhys. Lett. 26 (1994) 583 – 588. [12] P.A. Stadelmann, Ultramicroscopy 21 (2) (1987) 131 –145. [13] J.D. Rittner, D.N. Seidman, K.L. Merkle, Phys. Rev. B53 (1996) R4241 – R4244. [14] K.L. Merkle, Colloque Phys. C51 (1990) 251 – 256. [15] U. Wolf, F. Ernst, T. Muschik, M.W. Finnis, H.F. Fischmeister, Philos. Mag. A66 (1992) 991 –1016. [16] F. Ernst, M.W. Finnis, D. Hofmann, T. Muschik, U. Scho¨ nberger, U. Wolf, M. Methfessel, Phys. Rev. Lett. 69 (1992) 620 – 623.
[17] D.L. Medlin, G.H. Campbell, C. Barry Carter, Acta Mater. 46 (1998) 5135 – 5142. [18] B.J. Kooi, H.B. Groen, J.Th.M. De Hosson, Acta Mater. 45 (1997) 3587 – 3607. [19] J.Th.M. De Hosson, W.P. Vellinga, H.B. Groen, B.J. Kooi, in: D.J. Smith (Ed.), Atomic Resolution Microscopy of Surfaces and Interfaces, MRS Symp. Proc, vol. 466, Materials Research Society, Pittsburgh, PA, 1997, pp. 33 – 38. [20] J.P. Hirth, J. Lothe, Theory of Dislocations, second ed., Krieger Publishing Company, Malabar, FL, 1992, p. 324. [21] R.W. Balluffi, J. Nucl. Mater. 69&70 (1978) 240.
Structure and climb of 13 111 twin dislocations in aluminum S.M. Foiles a,*, D.L. Medlin b a
Sandia National Laboratories, Mail Stop 1411, P.O. Box 5800, Albuquerque, NM 87185, USA Sandia National Laboratories, Mail Stop 9161, P.O. Box 0969. Li6ermore, CA 94551, USA
b
Abstract The structure and motion via climb of 13 111 twin dislocations along a coherent twin boundary in aluminum are examined via a combination of high resolution transmission electron microscopy (HRTEM) and computer simulation. Climb of these defects increases the thickness of the twin and is a possible alternative to the more familiar glide mechanism. Detailed analysis of the observed defect structure and comparison with atomistic modeling shows that the 13 111 twin dislocations exist in a dissociated configuration. This structure can be interpreted as the relaxation of the 13 111 dislocation into two dislocations, a Shockley partial and a stair-rod dislocation, which are separated by a short segment of stacking fault. The atomistic mechanism of the climb of the dislocations is analyzed. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Structure; Twin dislocations; Aluminum
1. Introduction The formation of a twin boundary step by a gliding twin dislocation is well known (e.g. [1,2]). Perhaps less familiar is the formation of a FCC twin step by a dislocation of Burgers vector b = 13 111. This Burger’s vector is a member of the displacement shift complete (DSC) lattice [3], which represents the set of all translation vectors between the two perfect crystals. A schematic of this dislocation is shown in Fig. 1. This schematic illustrates that this dislocation corresponds to the termination of a single {111} plane of atoms. On the left side of the dislocation in Fig. 1, the {111} stacking sequence about the twin plane (indicated in bold face) is: CAB6 ACB. On the right side of the dislocation, the ‘A’ plane is removed giving a stacking sequence of CABC6 B, and thereby moving the twin boundary up by one (111) plane, producing a monolayer step. Direct, atomic scale evidence for such 1 3 111 dislocations at FCC twin boundaries has been obtained in high-resolution transmission electron microscopy (HRTEM) studies of aluminum [4–7] and gold [8]. * Corresponding author. Tel.: + 1-505-844-7064; fax: +1-505-8449781. E-mail address: [email protected] (S.M. Foiles).
Because the Burgers vector of the 13 111 dislocation is orthogonal to the twin plane, motion of this defect along the interface can occur only by climb. This is in contrast to the 16 [2( 11] dislocation, which moves by glide in the twin plane. We have recently reported in situ observations of such climb in an aluminum twin boundary under electron irradiation [5–7]. The structure of this boundary has been compared to atomistic simulation [7]. In the present paper, we review the detailed structure of the 13 111 dislocation and examine the energetics of vacancy formation in the boundary and the energetics associated with the climb process. We show that this defect relaxes significantly from the simple model depicted in Fig. 1, and that this relaxation can be understood in terms of a dislocation dissociation reaction. Further, the probable mechanism for the climb of the dislocation is examined. It is argued that the climb proceeds via a double kink formation process. 2. Observation of climbing 13 111 dislocations at a coherent twin interface The experimental observations presented here were made on a S= 3 (21( 1( )[01( 1] aluminum bicrystal, grown using a horizontal zone melting technique as described in Ref. [7]. This bicrystal was provided by C. Goux and
0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 1 ) 0 1 1 1 2 - 1
S.M. Foiles, D.L. Medlin / Materials Science and Engineering A319–321 (2001) 102–106
Fig. 1. Schematics showing the step in twin boundary produced by DSC lattice dislocation of Burgers vectors 13 [1( 1( 1( ]. Vertical lines of the same type are spaced at intervals of 16 [2( 11]. Horizontal lines are spaced at intervals of 13 [111]. Dashed lines are set into the plane of the figure by 14 [011( ].
M. Biscondi of the Ecole National Superieure des Mines, St. Etiennes, France. HRTEM observations of the boundary structure were conducted in a JEOL 4000EX electron microscope operated at 400 keV. As we have reported previously [5,7], a region of the S= 3 (21( 1( ) boundary roughened by the emission of narrow coherent twin lamellae during electron microscopic examination. Analysis of a sequence of micrographs taken during the growth of the lamella showed an array of steps associated with 13 111 dislocation on the top (111) facet that moved along the (111) facet during the growth of the twin. Emission and subsequent motion of the 13 111 dislocations along the coherent twin interface occurs by climb. The 400 keV electrons used for the observations are well above the 170 keV knock-on displacement threshold for Frenkel pair production in aluminum [9]. In general, such motion would also be conceivable under thermal processing especially at elevated temperatures. The experimental observations show that the structures relax significantly from the idealized model shown in Fig. 1. The {111} fringes near the step are offset such that the extra ‘half-fringe’ appears to be centered three (111) fringes above the twin plane. In contrast, the extra half-plane associated with the dislocation in the unrelaxed DSC lattice model is positioned immediately above the twin plane. The precise degree of this offset varies with offsets observed in the range of 2– 5 planes.
3. Interface modeling and comparison with HRTEM observations To better understand the interfacial dislocation relaxation, we have modeled the atomistic structure of a 1 3 111 twin dislocation using the embedded atom method [10,11]. The geometry of the calculations is described in detail elsewhere [7]. Fig. 2 shows a section of the relaxed structure in the vicinity of a 13 [1( 1( 1( ]
103
Fig. 2. Relaxed structure for 13 [1( 1( 1( ] interfacial dislocation at twin interface in Aluminum. The dashed line indicates the apparent position of the extra {111} plane and the lines labeled CT denote the position of the twin boundaries.
dislocation illustrating the resulting step in the coherent twin boundary (marked CT). The calculation predicts a relaxation from the compact structure of the DSC model: specifically, the {111} planes in the vicinity of the twin step are offset such that the extra {111} plane appears centered three planes above the coherent twin interface. The center of this distortion is marked on the calculated structure by a horizontal dashed line. This offset is consistent with the average observed experimentally. To compare the theoretical structure with the experimentally observed step configurations, we conducted HRTEM image simulations using the EMS package of electron microscopy simulation software [12]. Fig. 3a shows a simulated HRTEM image calculated from the theoretically derived atomic positions. Imaging parameters are summarized in the figure caption. An experimental image of a twin step and interfacial dislocation is shown in Fig. 3b overlaid with the positions of the intensity maximums from the theoretical image. Overall, there is a good, qualitative correspondence between the experimental and calculated images. An important feature to note in the calculated structure is the segment of stacking fault that extends from the twin step into the upper crystal along a (11( 1( ) plane. This short fault is marked on Fig. 2 by a black and white line and corresponds to a jog in the horizontal {111} planes. That interfaces may dissociate by the emission of stacking faults has been discussed recently [13], and a large number examples of such interfacial decomposition at interfaces in low stacking fault energy metals have now been reported [14–17]. Of particular relevance to the situation here are recent observations and calculations of 13 111 dislocations at tilted Ag/ Mn3O4 and Ag/ZnO interfaces, which were found to dissociate producing stacking faults extending from the interface into the metal [18,19].
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Fig. 3. Comparison of calculated structure for twin defect with experimental HRTEM observation. (a) Simulated HRTEM image for calculated defect structure shown in Fig. 5. Image simulation parameters: electron energy: 400 keV; spherical aberration coefficient: 1 mm; focus spread: 10 nm; semi-convergence angle: 1.0 mrad; specimen thickness: 5.7 nm; and defocus: 70 nm. (b) Experimental HRTEM image of twin defect shown in Fig. 3. Intensity peak positions from the simulated image shown in Fig. 6(a) are overlaid on the image.
Although aluminum has a high stacking fault energy a related dissociation (though to a more limited extent) has occurred here. A Frank dislocation can dissociate into a Shockley partial dislocation and stair-rod partial dislocation separated by a stacking fault (e.g. Ref. [20]). This dislocation reaction plays a key role in the formation of stacking fault tetrahedra in FCC metals of low stacking fault energy (SFE). As illustrated in Fig. 4, the perfect 13 [1( 1( 1( ] DSC dislocation (dD in Thompson’s notation) may also dissociate by emitting from the interface a Shockley partial dislocation (b= 16 [2( 1( 1( ] =bD) onto the (11( 1( ) plane. This produces a stacking fault on the (11( 1( ) plane and leaves behind a stair-rod dislocation (b = 16 [01( 1( ]= db) at the twin step. Referring to the calculated structure shown in Fig. 2, the core of the stair-rod dislocation (db) corresponds to the triangle of identically shaded atoms at the twin step and the stacking fault is indicated by the inclined solid line. The dissociated structure of the dislocation raises questions concerning the details of the climb process. In particular, the continuum picture suggests that the dissociated configuration must constrict in order to absorb vacancies and thereby climb. In order to address this issue, the energy and structure of various vacancy configurations has been examined. Fig. 5 presents a view of the formation energy of single vacancies in the vicinity of the dislocation core. The shading reflects local value of the vacancy formation energy. For the bulk crystal, the computed value of Efv is 0.69 eV. This compares with the experimental value of 0.67 eV [21]. The vacancy formation energy near the dislocation ranges from a low of 0.53 eV to a high of 0.91 eV. The low formation energies occur at the cores of both the stair-rod and Shockley par-
tials. Thus a single vacancy would tend to occupy either of those core positions with comparable frequency. The situation is substantially different when multiple vacancies move to the dislocation core. The formation energy for adding additional vacancies to vacancy clusters is plotted in Fig. 6a for clusters up to six vacancies. The qualitative behavior is quite different for the two cores. For the case of a vacancy cluster at the Shockley partial core, the formation energy saturates at a value of 0.26 eV. For the stairrod core, the formation energy for additional vacancies actually approaches zero. Thus it is energetically favorable for multiple vacancies to bind at the stairrod core. In order to understand the behavior of the formation energy of vacancy clusters at the stair rod core, consider the structure of the six vacancy cluster shown in Fig. 6b. The important point is that the local structure in the central region of this cluster is equivalent to that found for the straight dislocation. Thus the energy associated with the vacancy cluster corresponds to the energy to form the kinks at the
Fig. 4. Schematic showing geometry for dissociation of 13 [1( 1( 1( ] interfacial dislocation (dD) into a stair-rod dislocation (b = 16 [01( 1( ] = db), which remains at interface, and a Shockley partial (b = 16 [2( 1( 1( ] = bD), which leaves a stacking fault on the (11( 1( ) plane.
S.M. Foiles, D.L. Medlin / Materials Science and Engineering A319–321 (2001) 102–106
105
Fig. 5. Calculated values of the vacancy formation energies in the vicinity of the dislocation. The color represents the value of the vacancy formation energy with the lightest shade corresponding to 0.53 eV and the darkest shade to 0.91 eV.
edge of the cluster plus a small interaction energy between the kinks. Thus to add more vacancies to the cluster requires very little energy since this simply changes the separation of the kinks. Thus the climb of the dislocation is accomplished by a double kink mechanism. The main energy barrier is the initial formation of the double kink. The calculations also indicate that there is no energy barrier for the separation of the kinks as might be expected based on the dissociated nature of the core. Instead, the atomic structure relaxes without any barrier upon the addition of a vacancy.
4. Summary and concluding comments The presence of a 13 111 dislocation at a twin boundary produces a step that can, and does, move by climb. This is an important observation in that it provides a mechanism whereby twin growth can occur by non-conservative, thermally activated processes, rather than the conservative glide processes conventionally envisioned for twinning. The climb of the dislocation is seen to occur by a double kink type of mechanism. The formation of this double kink and its expansion does not appear to require an energy barrier. Such a barrier might be expected on continuum grounds if the dissociated dislocation needed to constrict in order to climb.
Acknowledgements This research is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under contract number DE-AC04-94-AL85000.
Fig. 6. (a) The vacancy formation energy in eV for the creation of the last vacancy of a cluster of size N. The triangles are for clusters at the Shockley core and the squares are at the stair-rod core. (b) The structure of a six vacancy cluster at the core of the stair-rod dislocation viewed along {111} directions. The shading represents the position normal the page. Note that the structure at the vacancy cluster is the same as for the straight boundary except at the two ends of the cluster.
References [1] F.C. Frank, J.H. van der Merwe, Proc. R. Soc. Lond. A198 (1949) 205 – 216. [2] W.T. Read, Dislocations in Crystals, McGraw-Hill, New York, 1953, p. 109. [3] W. Bollmann, Crystal Defects and Crystalline Interfaces, Springer-Verlag, New York, 1970. [4] M. Shamzuzzoha, P.A. Deymier, D.J. Smith, Philos. Mag. A64 (1991) 245 – 253. [5] D.L. Medlin, C.B. Carter, J.E. Angelo, M.J. Mills, Philos. Mag. A75 (1997) 733 – 747. [6] D.L. Medlin, S.M. Foiles, C.B. Carter, in: D.J. Smith (Ed.), Atomic Resolution Microscopy of Surfaces and Interfaces, MRS Symp. Proc, vol. 466, Materials Research Society, Pittsburgh, PA, 1997, pp. 125 – 130. [7] D.L. Medlin, in: S. Ankem, C.S. Pande (Eds.), Advances in Twinning, The Minerals, Metals and Materials Society, Warrendale, PA, 1999. [8] K.L. Merkle, J. Phys. Chem. Solids 55 (1994) 991 –1005. [9] A. Wolfenden, Radiat. Eff. 14 (1972) 225.
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[10] M.S. Daw, S.M. Foiles, M.I. Baskes, Mater. Sci. Rep. 9 (1993) 251. [11] F. Ercolessi, J.B. Adams, Europhys. Lett. 26 (1994) 583 – 588. [12] P.A. Stadelmann, Ultramicroscopy 21 (2) (1987) 131 –145. [13] J.D. Rittner, D.N. Seidman, K.L. Merkle, Phys. Rev. B53 (1996) R4241 – R4244. [14] K.L. Merkle, Colloque Phys. C51 (1990) 251 – 256. [15] U. Wolf, F. Ernst, T. Muschik, M.W. Finnis, H.F. Fischmeister, Philos. Mag. A66 (1992) 991 –1016. [16] F. Ernst, M.W. Finnis, D. Hofmann, T. Muschik, U. Scho¨ nberger, U. Wolf, M. Methfessel, Phys. Rev. Lett. 69 (1992) 620 – 623.
[17] D.L. Medlin, G.H. Campbell, C. Barry Carter, Acta Mater. 46 (1998) 5135 – 5142. [18] B.J. Kooi, H.B. Groen, J.Th.M. De Hosson, Acta Mater. 45 (1997) 3587 – 3607. [19] J.Th.M. De Hosson, W.P. Vellinga, H.B. Groen, B.J. Kooi, in: D.J. Smith (Ed.), Atomic Resolution Microscopy of Surfaces and Interfaces, MRS Symp. Proc, vol. 466, Materials Research Society, Pittsburgh, PA, 1997, pp. 33 – 38. [20] J.P. Hirth, J. Lothe, Theory of Dislocations, second ed., Krieger Publishing Company, Malabar, FL, 1992, p. 324. [21] R.W. Balluffi, J. Nucl. Mater. 69&70 (1978) 240.