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Lattice distortion and thermal stability of nano-crystalline copper
Computational Materials Science 30 (2004) 314–319 www.elsevier.com/locate/commatsci
Lattice distortion and thermal stability of nano-crystalline copper Lang Zhou *, Xiuqin Wei, Naigen Zhou School of Materials Science and Engineering, Nanchang University, Nanchang 330047, China Received 12 December 2003; received in revised form 18 February 2004; accepted 18 February 2004
Abstract Molecular dynamics simulations of high temperature annealing of copper bicrystals with varying grain sizes in nanometer range have been carried out. Planar h1 1 1i-tilt CSL grain boundaries are set. An EAM potential of FS type is used for calculating inter-atomic forces in copper. For comparison, similar simulations for aluminum and tungsten have been conducted. The results show that in the copper bicrystals of present grain boundary geometry, mismatch between the {1 1 1} planes of the neighboring grains occurs at the grain boundary, resulting in a general shear lattice distortion within the grains. The shear strain is inversely proportional to the grain size. The energy of such mismatched grain boundary is found lower than that of the mismatch-free grain boundary. For aluminum such kind of mismatch is much smaller, and for tungsten no such mismatch appears. The nano-sized copper bicrystals with grains smaller than a critical size are found instable at high temperature, where grain boundary motion and atomistic reconstruction lead to annihilation of the grain boundaries after an incubation time. 2004 Elsevier B.V. All rights reserved. PACS: 61.46; 71.15.D; 61.43.B; 61.72; 68.35.D; 68.60.D Keywords: Nano-crystalline metals; Lattice distortion; Grain boundary; Thermal stability; Molecular dynamics; Copper
1. Introduction Structure and thermal stability are natural concerns for nano-crystalline materials, especially for nano-metals. So far some achievements have been made to understand their structures [1]. Briefly, now we know that a nano-crystalline material consists of nano-sized crystals free of
*
Corresponding author. Tel.: +86-791-8304327; fax: +86791-8305141/8304441. E-mail address: [email protected] (L. Zhou).
dislocations, and grain boundaries of significant volume fraction. The structures of grain boundaries in different nano-metals are found to be either amorphous liquid-like, or the same as those in conventional polycrystalline materials. Since it has been established that grain boundaries in different metals of conventional grain size have generally similar structural feature, which is characterized by a rather ordered thin layer based on coincident lattice [2], the different observations for boundary structure of nano-grains of different metals seems to be contradictory and has caused argument. However, if we accept the complexity
0927-0256/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2004.02.013
L. Zhou et al. / Computational Materials Science 30 (2004) 314–319
that some unknown ‘‘minor’’ properties of a specific metal may affect its structural change in nanocrystallization, and hence not necessarily different metals have unique structural feature for their nano-crystalline grain boundaries, then the contradiction may not exist. The structures of the nano-crystals themselves, although free of local defects like dislocations [3,4], are found generally distorted in a way of isotropic lattice expansion [1,5,6]. No other type of distortion is reported up to now. In recent studies with molecular dynamics (MD) simulations, we have invariably found a shear distortion in nanobicrystal of copper with tilt grain boundary. Such distortion is then found to be much dependent on type of metals. The present paper reports these studies and further MD experimentations on thermal stability of the nano-crystalline copper. Molecular dynamics simulations can provide detailed structural information down to position and motion of each atom, and has been proven to be effective in understanding of grain boundary structure and its evolution [7–10]. It is further encouraging to notice that, in a few studies of GB structures [7,8], careful comparison of the results of using different potentials for inter-atomic reaction have confirmed that the structures are insensitive to details of the potentials.
315
boundaries. The tilt axis for the fcc metals is ½1 1 1. Fig. 1a serves as an illustration of the generated bicrystal cell, in which two crystallographic directions of the left grain are marked. For the bcc metal tungsten, the tilt axis is ½0 1 1, thus for both types of metals the concerned tilt grain boundaries have their tilt axis normal to the close packed atomic planes. The cells are 10 atomic layers thick and 36 atomic radii wide with each grain 18 atomic radii wide unless otherwise specified. Periodic boundary condition is applied to each dimension of the cell. Under such condition the left end and right end of the unit is joined and form another grain boundary. A periodic CSL grain boundary is applied to have perfect match of upper and lower
2. Methodology EAM potential of Finnis-Sinclair type for copper, developed by Auckland et al. [11], are adopted. It is fitted to copper’s lattice constant, cohesive energy, stacking fault energy and elastic moduli. The potential yields reasonably realistic values of vacancy formation energy and surface energy. The potentials for metals of reference, tungsten and aluminum, are of EAM type developed by Auckland and Thetford [12] and Ercolessi and Adams [13], respectively. Setup of the simulation cells are described as follows. Bicrystal cells with an asymmetrical R19 tilt grain boundary are generated with standard lattice constant and undistorted crystal geometry. Application of asymmetry and large CSL period number meant to represent random large angle grain
Fig. 1. Structure of high temperature annealed bicrystal copper. (a) ð1 1 1Þ projection, (b) ð1 1 2Þ projection, (c) ð1 1 2Þ projection, for the same type of bicrystal of double grain size.
316
L. Zhou et al. / Computational Materials Science 30 (2004) 314–319
ends of the cell in applying periodic border condition, so that no grain boundary or stacking fault forms at the periodic joints. For this purpose the height of the cell is set as the length of one structural period. The generated bicrystals are relaxed and annealed at 1000 K and zero pressure. Parrinello and Rahman’s constant stress algorithm [14] is adopted. The total energy and lateral size of the system are monitored throughout the simulation process. The length of time step is 1015 s. The error in calculation of the total energy is less than 104 eV. Snapshots of the atomic configuration of the system are taken from time to time to view structural evolution of the system. Before observation, the system is quenched to 0 K to clear thermal disturbance to coordinates of each atom.
Fig. 2. The concerned projections for the high temperature annealed bicrystals of different metals. (a) Aluminum at 700 K, (b) tungsten at 2500 K.
3. Results and discussion 3.1. Shear lattice distortion of the nano-bicrystals Fig. 1a and b show the structure of an annealed copper bicrystal cell in ð 1 1 1Þ projection and ð1 1 2Þ projection, respectively. Fig. 1a gives a general image of the structure obtained. Fig. 1b reveals a shear distortion of the whole grains resulting from mismatch of each pair of ð 1 1 1Þ planes of the two neighboring grains. Instead of meeting on the same plane, they intersect with each other, with a plane of one grain matching the centre of the interplane space of the other. Fig. 1c shows such projection for the same type of bicrystal with doubled grain size. It is explicitly seen that d remains unchanged, and the shear strain c in a grain is inversely proportional to its width, i.e., c ¼ 2d=d, where d represents the distance of mismatch, which is half of the inter-plane distance here, and d the width of the grain. So, this shear distortion is negligible for grains of conventional size, but significant for nano-sized grains. Similar simulation experiments have been carried out for an alternative fcc metal, aluminum, and a bcc metal, tungsten. Fig. 2 displays the concerned projections of the relaxed cells. It is seen that the general shear distortion also occurs in aluminum, though less remarkable with smaller d,
which is only about 1/8 of the inter-plane distance, while for tungsten, there is no such kind of distortion at all. The reason for such difference remains to be sought. However, at present stage we can see that in nano-crystallization, large variation in structural feature may arise for different metals, even though they may have similar structural feature in conventional polycrystals. Relaxation at 0 K for the bicrystal cells generated in the same way has also been carried out for copper. The shear distortion does not appear here. The energies of these cells are compared with the energies of those bearing the shear distortion at 0 K, as shown in Table 1. The cells of the latter were annealed at 1000 K for 105 steps, gradually cooled to 20 K in about 106 steps, and then quenched to 0 K, where the shear distortion was found to remain. Energies for perfect crystals consisting of the same amount of atoms, and for the perfect crystal applied with the same shear distortion as appeared in the bicrystal, and the deduced grain boundary energies, along with experimentally determined average grain boundary energy for conventional polycrystalline copper [15], are also listed in Table 1 for reference. The grain boundary energy is calculated by ðEbicrystal Eperfect Þ=AGB , where Ebicrystal and Eperfect stand for the total energies of the bicrystal cell and a perfect crystal cell, and AGB
L. Zhou et al. / Computational Materials Science 30 (2004) 314–319
317
Table 1 Energies of the bicrystal and single crystal cells (5472 atoms) with and without shear lattice distortion and grain boundary energies at 0 K Bicrystal (eV)
GB energy (J/m2 )
GB energy measured by experiment [15] (J/m2 )
)19258.42 )19231.52 26.90
)19044.44 )19039.97 4.47
1.08 0.97
0.63 (at 950 C)
the total grain boundary area. It is emphasized that for the bicrystal bearing shear lattice distortion, the total energy of the perfect crystal with the same shear strain is used for Eperfect , so that the strain energy is not counted into the grain boundary energy. We notice that although the shear strain energy shown in the perfect crystal calculation is as high as 26.90 eV, the total energy of the bicrystal with the shear distortion is only 4.47 eV higher than that of the bicrystal without shear distortion. Even though about 20% atoms are in grain boundaries, not involved in the lattice shear distortion, 4.47 eV is still far from enough to account for the 80% of the strain energy. The only possibility is that the structural reconstruction at high temperature, which leads to the observed mismatch at grain boundaries, has significantly lowered the grain boundary energy. The calculation shows that it is about 10% lower. At high temperature, the elastic moduli, and hence the strain energy, is significantly lower than that at 0 K. So it is possible that the total energy of the bicrystal with the shear distortion associated with the mismatch at grain boundary becomes lower than that without the shear distortion, and hence it becomes energetically favorable, at high temperature. In cooling and quench to 0 K, the mismatch maintains because of lack of time for reconstruction, though the total energy is slightly higher than that without shear distortion. Due to the disturbance from vibration of atoms at high temperature, accurate calculation to verify the above postulation is not possible yet. 3.2. Thermal stability of the nano-bicrystals Simulations of prolonged high temperature annealing for the bicrystals of smaller grain size have been carried out. Fig. 3 shows the evolution
-15900
Normalized Total Potential Energy
c¼0 c ¼ 0:045 Ec¼0:045 Ec¼0
Perfect crystal (eV)
-15950
-16000
A B
-16050
d=16r
C
-16100
d=14r d=12r
-16150 0
2 00
400
600
800
Time (PS)
Fig. 3. The total potential energies of the bicrystals with various grain sizes in the isothermal annealing at 1000 K. Snapshots of the structures taken at points A, B, C are displayed in Fig. 4.
of the total potential energy of the bicrystal cells with different grain size, d, in the simulation processes. The energy value is normalized to represent the same total number of atoms as that of the cell with d ¼ 14r, where r is the atomic radii. The associated structural evolution is demonstrated in Fig. 4. Note that with periodic borders, the rightward expansion of the right grain extends to the left end of the cell. In the case of d ¼ 16r, no significant structural change occurred within the maximum time of running, 2 · 106 steps, and the total energy of the system remains at the same level, as can be seen partially in Fig. 3. However, when the grain size becomes smaller, rapid drop of energy occurs at a certain stage, accompanied by grain boundary migration and complex structural reconstructions, and finally annihilation of the grain boundary, as exhibited in Fig. 4a–c. Note that the original position of the grain boundary is at the centre of the bicrystal. It is interesting to see
318
L. Zhou et al. / Computational Materials Science 30 (2004) 314–319
fluctuation [2] may have induced the process. As a result, the sizes of the two grains become different, and a shear strain energy contrast between grains of different size appears. This energy contrast should provide the driving force for further motion of the grain boundary, which in turn raises the energy contrast. In addition, interaction between the neighboring grain boundaries may also be a driving force for occurrence of the structural instability [16]. Both kinds of the driving force weaken rapidly with increase of grain size, leading to the existence of a critical size for occurring of the instability. 4. Concluding remark
Fig. 4. Snapshots of ð1 1 1Þ projections of the bicrystal with d ¼ 14r, taken at the points labeled in Fig. 3. (a) Point A in Fig. 3, at 320 ps, (b) point B in Fig. 3, at 360 ps, (c) point C in Fig. 3, at 500 ps.
that, in the brief transition, formation of an intermediate grain in between the two original grains is involved, as appears in the lower left part of Fig. 4a, and it soon disappears. Significance of this phenomenon to practical aspects of nanocrystals’ instability is to be addressed in future. The results also indicate that before the transition starts, an incubation time is required, which is strongly size dependent. The grain width of 16r seems to be a critical size, beyond which the bicrystal remains stable, and just one atomic diameter below which, the bicrystal becomes instable in short time. As planar grain boundaries provide no driving force for grain boundary motion, it is suggested that drift motion of the grain boundary by thermal
A shear lattice distortion in nano-crystalline copper with a h1 1 1i-tilt grain boundary, associated with mismatch of {1 1 1} planes of the neighboring grains at the grain boundary, has been identified. The shear strain simply equals to 2d=d, where d is the distance of the mismatch and d the grain size. The mismatch distance d is found independent of grain size, while much dependant on type of metals. For copper, d is half of the interplane distance, for aluminum, d is much smaller, while for tungsten it is zero. Energy of the copper grain boundary with the mismatch is found lower than that of the grain boundary without such mismatch. Grain boundary motion and atomistic reconstruction leading to annihilation of the grain boundaries occurs for copper bicrystals with grain size below a critical size. In the present case the critical size is found to be sixteen atomic radii. Acknowledgements Support of the present work by the Natural Science Foundation of China under grant no. 20021024 is gratefully acknowledged. References [1] H. Gleiter, Acta Mater. 48 (2000) 1–29. [2] G. Gottstein, L.S. Shvindlerman, Grain Boundary Migration in Metals, CRC Press, London, 1999.
L. Zhou et al. / Computational Materials Science 30 (2004) 314–319 [3] J. Schiotz, F.D. Di Tolla, K.W. Jacobsen, Nature 391 (1998) 561–563. [4] H. Van Swygenhoven, A. Caro, Appl. Phys. Lett. 71 (1997) 1652–1654. [5] D. Chen, Comput. Mater. Sci. 3 (1995) 327–333. [6] B. Peng, M. Cai, G. Li, X.J. Wu, F. Zhou, Nanostruct. Mater. 4 (1994) 475–484. [7] R.J. Harrison, G.A. Bruggeman, G.H. Bishop, in: G.A. Chadwick, D.A. Smith (Eds.), Grain Boundary and Properties, Academic Press, London, 1976, pp. 45–86. [8] H. van Swygenhoven, Phys. Rev. 62 (2000) 831–840. [9] B. Schonfelder, D. Wolf, S.R. Phillpot, M. Furtkamp, Interf. Sci. 5 (1997) 245–262.
319
[10] A.J. Haslam, D. Moldovan, S.R. Phillpot, D. Wolf, Comput. Mater. Sci. 23 (2002) 15–25. [11] G.J. Auckland, G. Tichy, V. Vitek, M.W. Finnis, Philos. Mag. 56A (1987) 735–748. [12] G.J. Auckland, R. Thetford, Philos. Mag. 56A (1987) 15– 26. [13] F. Ercolessi, J. Adams, Europhys. Lett. 26 (1994) 583– 592. [14] M. Parrinello, A. Rahman, J. Appl. Phys. 52 (1981) 7182– 7188. [15] J.M. Howe, Interfaces in Materials, John Wiley & Sons, Inc., New York, 1997, p. 308. [16] L. Zhou, S. Zhu, Scr. Mater. 47 (2002) 677–682.
Lattice distortion and thermal stability of nano-crystalline copper Lang Zhou *, Xiuqin Wei, Naigen Zhou School of Materials Science and Engineering, Nanchang University, Nanchang 330047, China Received 12 December 2003; received in revised form 18 February 2004; accepted 18 February 2004
Abstract Molecular dynamics simulations of high temperature annealing of copper bicrystals with varying grain sizes in nanometer range have been carried out. Planar h1 1 1i-tilt CSL grain boundaries are set. An EAM potential of FS type is used for calculating inter-atomic forces in copper. For comparison, similar simulations for aluminum and tungsten have been conducted. The results show that in the copper bicrystals of present grain boundary geometry, mismatch between the {1 1 1} planes of the neighboring grains occurs at the grain boundary, resulting in a general shear lattice distortion within the grains. The shear strain is inversely proportional to the grain size. The energy of such mismatched grain boundary is found lower than that of the mismatch-free grain boundary. For aluminum such kind of mismatch is much smaller, and for tungsten no such mismatch appears. The nano-sized copper bicrystals with grains smaller than a critical size are found instable at high temperature, where grain boundary motion and atomistic reconstruction lead to annihilation of the grain boundaries after an incubation time. 2004 Elsevier B.V. All rights reserved. PACS: 61.46; 71.15.D; 61.43.B; 61.72; 68.35.D; 68.60.D Keywords: Nano-crystalline metals; Lattice distortion; Grain boundary; Thermal stability; Molecular dynamics; Copper
1. Introduction Structure and thermal stability are natural concerns for nano-crystalline materials, especially for nano-metals. So far some achievements have been made to understand their structures [1]. Briefly, now we know that a nano-crystalline material consists of nano-sized crystals free of
*
Corresponding author. Tel.: +86-791-8304327; fax: +86791-8305141/8304441. E-mail address: [email protected] (L. Zhou).
dislocations, and grain boundaries of significant volume fraction. The structures of grain boundaries in different nano-metals are found to be either amorphous liquid-like, or the same as those in conventional polycrystalline materials. Since it has been established that grain boundaries in different metals of conventional grain size have generally similar structural feature, which is characterized by a rather ordered thin layer based on coincident lattice [2], the different observations for boundary structure of nano-grains of different metals seems to be contradictory and has caused argument. However, if we accept the complexity
0927-0256/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2004.02.013
L. Zhou et al. / Computational Materials Science 30 (2004) 314–319
that some unknown ‘‘minor’’ properties of a specific metal may affect its structural change in nanocrystallization, and hence not necessarily different metals have unique structural feature for their nano-crystalline grain boundaries, then the contradiction may not exist. The structures of the nano-crystals themselves, although free of local defects like dislocations [3,4], are found generally distorted in a way of isotropic lattice expansion [1,5,6]. No other type of distortion is reported up to now. In recent studies with molecular dynamics (MD) simulations, we have invariably found a shear distortion in nanobicrystal of copper with tilt grain boundary. Such distortion is then found to be much dependent on type of metals. The present paper reports these studies and further MD experimentations on thermal stability of the nano-crystalline copper. Molecular dynamics simulations can provide detailed structural information down to position and motion of each atom, and has been proven to be effective in understanding of grain boundary structure and its evolution [7–10]. It is further encouraging to notice that, in a few studies of GB structures [7,8], careful comparison of the results of using different potentials for inter-atomic reaction have confirmed that the structures are insensitive to details of the potentials.
315
boundaries. The tilt axis for the fcc metals is ½1 1 1. Fig. 1a serves as an illustration of the generated bicrystal cell, in which two crystallographic directions of the left grain are marked. For the bcc metal tungsten, the tilt axis is ½0 1 1, thus for both types of metals the concerned tilt grain boundaries have their tilt axis normal to the close packed atomic planes. The cells are 10 atomic layers thick and 36 atomic radii wide with each grain 18 atomic radii wide unless otherwise specified. Periodic boundary condition is applied to each dimension of the cell. Under such condition the left end and right end of the unit is joined and form another grain boundary. A periodic CSL grain boundary is applied to have perfect match of upper and lower
2. Methodology EAM potential of Finnis-Sinclair type for copper, developed by Auckland et al. [11], are adopted. It is fitted to copper’s lattice constant, cohesive energy, stacking fault energy and elastic moduli. The potential yields reasonably realistic values of vacancy formation energy and surface energy. The potentials for metals of reference, tungsten and aluminum, are of EAM type developed by Auckland and Thetford [12] and Ercolessi and Adams [13], respectively. Setup of the simulation cells are described as follows. Bicrystal cells with an asymmetrical R19 tilt grain boundary are generated with standard lattice constant and undistorted crystal geometry. Application of asymmetry and large CSL period number meant to represent random large angle grain
Fig. 1. Structure of high temperature annealed bicrystal copper. (a) ð1 1 1Þ projection, (b) ð1 1 2Þ projection, (c) ð1 1 2Þ projection, for the same type of bicrystal of double grain size.
316
L. Zhou et al. / Computational Materials Science 30 (2004) 314–319
ends of the cell in applying periodic border condition, so that no grain boundary or stacking fault forms at the periodic joints. For this purpose the height of the cell is set as the length of one structural period. The generated bicrystals are relaxed and annealed at 1000 K and zero pressure. Parrinello and Rahman’s constant stress algorithm [14] is adopted. The total energy and lateral size of the system are monitored throughout the simulation process. The length of time step is 1015 s. The error in calculation of the total energy is less than 104 eV. Snapshots of the atomic configuration of the system are taken from time to time to view structural evolution of the system. Before observation, the system is quenched to 0 K to clear thermal disturbance to coordinates of each atom.
Fig. 2. The concerned projections for the high temperature annealed bicrystals of different metals. (a) Aluminum at 700 K, (b) tungsten at 2500 K.
3. Results and discussion 3.1. Shear lattice distortion of the nano-bicrystals Fig. 1a and b show the structure of an annealed copper bicrystal cell in ð 1 1 1Þ projection and ð1 1 2Þ projection, respectively. Fig. 1a gives a general image of the structure obtained. Fig. 1b reveals a shear distortion of the whole grains resulting from mismatch of each pair of ð 1 1 1Þ planes of the two neighboring grains. Instead of meeting on the same plane, they intersect with each other, with a plane of one grain matching the centre of the interplane space of the other. Fig. 1c shows such projection for the same type of bicrystal with doubled grain size. It is explicitly seen that d remains unchanged, and the shear strain c in a grain is inversely proportional to its width, i.e., c ¼ 2d=d, where d represents the distance of mismatch, which is half of the inter-plane distance here, and d the width of the grain. So, this shear distortion is negligible for grains of conventional size, but significant for nano-sized grains. Similar simulation experiments have been carried out for an alternative fcc metal, aluminum, and a bcc metal, tungsten. Fig. 2 displays the concerned projections of the relaxed cells. It is seen that the general shear distortion also occurs in aluminum, though less remarkable with smaller d,
which is only about 1/8 of the inter-plane distance, while for tungsten, there is no such kind of distortion at all. The reason for such difference remains to be sought. However, at present stage we can see that in nano-crystallization, large variation in structural feature may arise for different metals, even though they may have similar structural feature in conventional polycrystals. Relaxation at 0 K for the bicrystal cells generated in the same way has also been carried out for copper. The shear distortion does not appear here. The energies of these cells are compared with the energies of those bearing the shear distortion at 0 K, as shown in Table 1. The cells of the latter were annealed at 1000 K for 105 steps, gradually cooled to 20 K in about 106 steps, and then quenched to 0 K, where the shear distortion was found to remain. Energies for perfect crystals consisting of the same amount of atoms, and for the perfect crystal applied with the same shear distortion as appeared in the bicrystal, and the deduced grain boundary energies, along with experimentally determined average grain boundary energy for conventional polycrystalline copper [15], are also listed in Table 1 for reference. The grain boundary energy is calculated by ðEbicrystal Eperfect Þ=AGB , where Ebicrystal and Eperfect stand for the total energies of the bicrystal cell and a perfect crystal cell, and AGB
L. Zhou et al. / Computational Materials Science 30 (2004) 314–319
317
Table 1 Energies of the bicrystal and single crystal cells (5472 atoms) with and without shear lattice distortion and grain boundary energies at 0 K Bicrystal (eV)
GB energy (J/m2 )
GB energy measured by experiment [15] (J/m2 )
)19258.42 )19231.52 26.90
)19044.44 )19039.97 4.47
1.08 0.97
0.63 (at 950 C)
the total grain boundary area. It is emphasized that for the bicrystal bearing shear lattice distortion, the total energy of the perfect crystal with the same shear strain is used for Eperfect , so that the strain energy is not counted into the grain boundary energy. We notice that although the shear strain energy shown in the perfect crystal calculation is as high as 26.90 eV, the total energy of the bicrystal with the shear distortion is only 4.47 eV higher than that of the bicrystal without shear distortion. Even though about 20% atoms are in grain boundaries, not involved in the lattice shear distortion, 4.47 eV is still far from enough to account for the 80% of the strain energy. The only possibility is that the structural reconstruction at high temperature, which leads to the observed mismatch at grain boundaries, has significantly lowered the grain boundary energy. The calculation shows that it is about 10% lower. At high temperature, the elastic moduli, and hence the strain energy, is significantly lower than that at 0 K. So it is possible that the total energy of the bicrystal with the shear distortion associated with the mismatch at grain boundary becomes lower than that without the shear distortion, and hence it becomes energetically favorable, at high temperature. In cooling and quench to 0 K, the mismatch maintains because of lack of time for reconstruction, though the total energy is slightly higher than that without shear distortion. Due to the disturbance from vibration of atoms at high temperature, accurate calculation to verify the above postulation is not possible yet. 3.2. Thermal stability of the nano-bicrystals Simulations of prolonged high temperature annealing for the bicrystals of smaller grain size have been carried out. Fig. 3 shows the evolution
-15900
Normalized Total Potential Energy
c¼0 c ¼ 0:045 Ec¼0:045 Ec¼0
Perfect crystal (eV)
-15950
-16000
A B
-16050
d=16r
C
-16100
d=14r d=12r
-16150 0
2 00
400
600
800
Time (PS)
Fig. 3. The total potential energies of the bicrystals with various grain sizes in the isothermal annealing at 1000 K. Snapshots of the structures taken at points A, B, C are displayed in Fig. 4.
of the total potential energy of the bicrystal cells with different grain size, d, in the simulation processes. The energy value is normalized to represent the same total number of atoms as that of the cell with d ¼ 14r, where r is the atomic radii. The associated structural evolution is demonstrated in Fig. 4. Note that with periodic borders, the rightward expansion of the right grain extends to the left end of the cell. In the case of d ¼ 16r, no significant structural change occurred within the maximum time of running, 2 · 106 steps, and the total energy of the system remains at the same level, as can be seen partially in Fig. 3. However, when the grain size becomes smaller, rapid drop of energy occurs at a certain stage, accompanied by grain boundary migration and complex structural reconstructions, and finally annihilation of the grain boundary, as exhibited in Fig. 4a–c. Note that the original position of the grain boundary is at the centre of the bicrystal. It is interesting to see
318
L. Zhou et al. / Computational Materials Science 30 (2004) 314–319
fluctuation [2] may have induced the process. As a result, the sizes of the two grains become different, and a shear strain energy contrast between grains of different size appears. This energy contrast should provide the driving force for further motion of the grain boundary, which in turn raises the energy contrast. In addition, interaction between the neighboring grain boundaries may also be a driving force for occurrence of the structural instability [16]. Both kinds of the driving force weaken rapidly with increase of grain size, leading to the existence of a critical size for occurring of the instability. 4. Concluding remark
Fig. 4. Snapshots of ð1 1 1Þ projections of the bicrystal with d ¼ 14r, taken at the points labeled in Fig. 3. (a) Point A in Fig. 3, at 320 ps, (b) point B in Fig. 3, at 360 ps, (c) point C in Fig. 3, at 500 ps.
that, in the brief transition, formation of an intermediate grain in between the two original grains is involved, as appears in the lower left part of Fig. 4a, and it soon disappears. Significance of this phenomenon to practical aspects of nanocrystals’ instability is to be addressed in future. The results also indicate that before the transition starts, an incubation time is required, which is strongly size dependent. The grain width of 16r seems to be a critical size, beyond which the bicrystal remains stable, and just one atomic diameter below which, the bicrystal becomes instable in short time. As planar grain boundaries provide no driving force for grain boundary motion, it is suggested that drift motion of the grain boundary by thermal
A shear lattice distortion in nano-crystalline copper with a h1 1 1i-tilt grain boundary, associated with mismatch of {1 1 1} planes of the neighboring grains at the grain boundary, has been identified. The shear strain simply equals to 2d=d, where d is the distance of the mismatch and d the grain size. The mismatch distance d is found independent of grain size, while much dependant on type of metals. For copper, d is half of the interplane distance, for aluminum, d is much smaller, while for tungsten it is zero. Energy of the copper grain boundary with the mismatch is found lower than that of the grain boundary without such mismatch. Grain boundary motion and atomistic reconstruction leading to annihilation of the grain boundaries occurs for copper bicrystals with grain size below a critical size. In the present case the critical size is found to be sixteen atomic radii. Acknowledgements Support of the present work by the Natural Science Foundation of China under grant no. 20021024 is gratefully acknowledged. References [1] H. Gleiter, Acta Mater. 48 (2000) 1–29. [2] G. Gottstein, L.S. Shvindlerman, Grain Boundary Migration in Metals, CRC Press, London, 1999.
L. Zhou et al. / Computational Materials Science 30 (2004) 314–319 [3] J. Schiotz, F.D. Di Tolla, K.W. Jacobsen, Nature 391 (1998) 561–563. [4] H. Van Swygenhoven, A. Caro, Appl. Phys. Lett. 71 (1997) 1652–1654. [5] D. Chen, Comput. Mater. Sci. 3 (1995) 327–333. [6] B. Peng, M. Cai, G. Li, X.J. Wu, F. Zhou, Nanostruct. Mater. 4 (1994) 475–484. [7] R.J. Harrison, G.A. Bruggeman, G.H. Bishop, in: G.A. Chadwick, D.A. Smith (Eds.), Grain Boundary and Properties, Academic Press, London, 1976, pp. 45–86. [8] H. van Swygenhoven, Phys. Rev. 62 (2000) 831–840. [9] B. Schonfelder, D. Wolf, S.R. Phillpot, M. Furtkamp, Interf. Sci. 5 (1997) 245–262.
319
[10] A.J. Haslam, D. Moldovan, S.R. Phillpot, D. Wolf, Comput. Mater. Sci. 23 (2002) 15–25. [11] G.J. Auckland, G. Tichy, V. Vitek, M.W. Finnis, Philos. Mag. 56A (1987) 735–748. [12] G.J. Auckland, R. Thetford, Philos. Mag. 56A (1987) 15– 26. [13] F. Ercolessi, J. Adams, Europhys. Lett. 26 (1994) 583– 592. [14] M. Parrinello, A. Rahman, J. Appl. Phys. 52 (1981) 7182– 7188. [15] J.M. Howe, Interfaces in Materials, John Wiley & Sons, Inc., New York, 1997, p. 308. [16] L. Zhou, S. Zhu, Scr. Mater. 47 (2002) 677–682.