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Molecular dynamics simulations of dislocations and nanocrystals
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Current Applied Physics 8 (2008) 494–497 www.elsevier.com/locate/cap www.kps.or.kr
Molecular dynamics simulations of dislocations and nanocrystals David Stewart a,*, Ke-Shen Cheong b a
Industrial Research Limited, Gracefield Research Centre, P.O. Box 31-310, Lower Hutt 5040, New Zealand b MPT Solutions Limited, Gracefield Research Centre, P.O. Box 31-310, Lower Hutt 5040, New Zealand Available online 26 October 2007
Abstract Molecular dynamics simulations are used to investigate the mechanical properties of FCC aluminium at the atomic level. As the MD method involves integrating the equations of motion to obtain trajectories for all the particles in the simulation, it is very general. Dislocation–dislocation interactions are simulated to try to find the critical distance at which opposite sign dislocations will mutually annihilate. Simulations of the interaction of a dislocation with a void show that the attraction slows down the dislocation motion, leading to void strengthening. Simulations of the deformation of an aluminium polycrystal with a grain size of 8.6 nm show that not just dislocation motion but also twinning plays a significant role. Twinning is possible in nanocrystalline aluminium because the material can withstand sufficient stress to force partial dislocations separation comparable to the grain size. Ó 2007 Published by Elsevier B.V. PACS: 61.72.Ji; 61.72.Lk; 61.72.Mm; 61.72.Nn; 61.82.Rx; 62.20.x; 62.20.Fe; 82.20.Wt Keywords: Molecular dynamics; Dislocations; Voids; Nanocrystal; Stacking fault; Twinning; Al
1. Introduction Molecular dynamics (MD) is a useful tool for investigating mechanical properties at the nanoscale. The most important deformation mechanism is the motion of dislocations. Dislocation interactions depend on the details of the dislocation core and thus atomistic simulations provide better accuracy than continuum modelling. However, atomistic methods are much more computationally expensive, and thus are limited to smaller regions and shorter timescales. Cheong and Busso’s plasticity model [1–3] predicts the deformation behaviour of polycrystal aggregates by simulating the evolution of densities of edge and screw dislocations in the material. This evolution depends on the distance at which opposite sign dislocations on the same slip system will be able to mutually annihilate. The values for these distances are inferred from temperature-cycling
*
Corresponding author. Tel.: +64 4 931 3715; fax: +64 4 931 3003. E-mail address: [email protected] (D. Stewart).
1567-1739/$ - see front matter Ó 2007 Published by Elsevier B.V. doi:10.1016/j.cap.2007.10.044
experiments. MD simulations offer a way to determine these distances in a more direct way. Irradiated materials contain voids and other obstacles to dislocation glide that can alter the mechanical properties. MD is a good candidate for modelling the dislocation–void interaction, as it depends on the dislocation core structure [4–6]. Nanocrystalline (nc) materials are of interest because they have many properties that are significantly different from the equivalent coarse-grained materials [7,8]. The Hall-Petch relationship states that the strength varies as (grain size)1/2. Nanocrystalline aluminium has been experimentally found to follow this relationship at least down to 15 nm [9,10] and to become more wear resistant with decreasing grain size [10]. The Hall–Petch effect seen in coarse-grained aluminium is a result of dislocation motion. MD simulations on nc Al [13–14] have found deformation by grain-boundary sliding and twinning [13,14], leading to a breakdown of the HP effect. In this paper, we investigate some of the properties of dislocations using MD simulations. First we look at the possible annihilation of opposite dislocations. Then, the
D. Stewart, K.-S. Cheong / Current Applied Physics 8 (2008) 494–497
495
interaction of a dislocation with a void. Then we look at the mechanisms that appear when a nc material is deformed. 2. Methodology
3. Dislocation annihilation We have simulated the interaction of dislocations in aluminium. Two screw dislocations with Burger’s vector 1/2 ½1 1 0 are placed in an FCC crystal of dimensions 35 ½1 1 0 by 30 [1 1 1] by 40 ½1 1 2. The dislocations are in glide planes separated by a number of [1 1 1] planes. 50 MPa of external shear stress is applied to the system. Simulations are done for several separation distances. Fig. 1 shows a simulation where the glide planes are separated by 413 [1 1 1], i.e. 13 atom planes. For screw dislocations at 200 K, annihilation is seen if the number of [1 1 1] planes is 13 or less. If the distance is larger than this, they simply pass by. 4. Dislocation–void interactions We have also simulated the interaction of dislocations with voids in aluminium. Two screw dislocations with Burger’s vector 1/2 ½1 1 0 are placed in an FCC crystal of dimensions 70 ½1 1 0 by 30 [1 1 1] by 20 ½1 1 2. A spherical void with diameter 4 nm is placed in the path of one of the dislocations, with its centre in the glide plane. 50 MPa of external shear stress is applied to the system. The dislocation travels at approximately 0.1 m/s through the crystal. The dislocation bends as it breaks away from the void, as shown in Fig. 2. The dislocation takes an extra 5 ps to get past the void.
Fig. 2. Screw dislocation initially (left) and interacting with a 4 nm void (right) at 20 ps.
5. Nanocrystal deformation A Voronoi construction is used to generate a periodic cell containing 24 grains. For each grain, a random location in the cell is chosen as the grain centre. Then, any point in the cell is considered part of the grain whose centre is closest. The 490,083-atom cell is approximately 20 nm cubed and the grains range in size from 6.8 nm (10,100 atoms) to 10.1 nm (32,708 atoms) with an average of 8.6 nm. Each grain is filled with a randomly orientated FCC lattice. A gap of 0.25 a0 (where a0 is the lattice parameter) is left between grains to avoid generating atoms closer than nearest neighbour distance apart. The cell is equilibrated for a nanosecond of simulation time in order to allow the system to relax into a steady state. The result is a nanocrystal with a range of grain shapes and sizes and a range of different grain boundaries. The cell is then sheared at a constant deformation rate at 200 K. Fig. 3 shows the stress-strain curves for deformations of 5 108 and 5 107 strains per second. As the strain on the nanocrystal is increased, we have observed several different deformation processes. In the initial part of Fig. 3, where the stress is monotonically increasing, dislocations are not obvious; most deformation is by grain boundary sliding as seen in Ref. [11]. In the upper grain shown in Fig. 4, a partial dislocation has nucleated at a grain boundary and crossed the grain, Stress vs Strain 1.4 1.2 1
Stress (GPa)
Our MD simulations use periodic boundary conditions with a Ray–Rahmen flexible simulation cell [15] in order to allow the cell to change size and shape in response to stress. The equations of motion are integrated using a Gear predictor–corrector algorithm [16]. A Nose´-Hoover thermostat is used to control the temperature. The simulations are run in parallel using a domain decomposition algorithm [17]. The Ercolessi-Adams embedded atom potential for aluminium is used [18]. This potential has a stacking fault energy of 104 mJ/m2, compared to experimental measurements of 120–142 mJ/m2 [19].
0.8 0.6 8
Strain rate 5x10 s–1 7 Strain rate 5x10 s–1
0.4 0.2 0 Fig. 1. Periodic cell with two screw dislocations in their initial positions (left) and as they annihilate after 16 ps (right). The glide planes are separated by 13 atomic planes (3.0 nm). Created using atomeye [26].
0
0.02
0.04
0.06
0.08
0.1
Strain Fig. 3. Stress–strain curves for an aluminium Voronoi 8.6 nm nanocrystal deformed at two different strain rates at 200 K.
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D. Stewart, K.-S. Cheong / Current Applied Physics 8 (2008) 494–497
Fig. 4. Voronoi nanocrystal after 8.3% strain. Right image is a close up of the left image. The upper grain shows twinning – the light atoms show the twin planes. The light atoms in the lower grain are a screw dislocation. Created using atomeye [26]. See appendix for details of atom colouring.
creating a stacking fault in its wake. This stacking fault then nucleated twinning. A stacking fault is equivalent to two twinning planes next to each other. These planes move apart via the motion of more partial dislocations, creating a grain between them. The orientation of the new grain is related by mirror symmetry. In the lower grain in Fig. 4, two partial dislocations nucleated and then moved together, creating a full screw dislocation. We have observed cross-slip of screw dislocations that were created this way. 6. Discussion Twinning is not seen in coarse-grained aluminium due to the high stacking fault energy. When a full 1/2 h1 1 0i dislocation dissociates into two 1/6 h1 1 2i partial dislocations, the repulsive force between the partials is balanced by the stacking fault energy, leading to an equilibrium separation distance [23,24]. The Burger’s vectors of the partials differ by 60° and therefore respond to stress differently. This leads to an extra force between the partials proportional to rDb [12,24]. If the resolved shear stress in the glide plane is repulsive and greater than 2c/bp (1.3 GPa for the potential in Ref. [18]), it completely outweighs the attraction; the partials can always lower their energy by moving further apart [12,25]. This level of stress is only possible in the nanocrystalline case, as coarse-grained Al would yield well below this stress level. Even though the choice of mechanism is stress-dependant, both mechanisms were seen at the same level of applied stress. Because the grains are oriented differently, the resolved stress on glide planes in different grains is different. 7. Future work We have simulated dislocation–dislocation and dislocation–void interactions in aluminium. The periodic boundary conditions mean that the dislocation pair is effectively in an infinite lattice of pairs. The elastic interactions between them are long range and lead to
stress variations in the primary cell that alter the result. [20–22]. Ref. [20] shows that for BCC, these spurious effects can be cancelled out if a special aspect ratio is used. More work is needed to clarify the situation for FCC. This will allow the calculation of the dislocation–dislocation annihilation distance and the critical resolved shear stress required for a dislocation to break away from a void-parameters that can be fed into the model of Cheong & Busso. We have simulated the deformation of a nanocrystalline material and observed deformation by twinning and by full dislocations. The effects of grain size and temperature can be investigated. We use the well-established potential from Ref. [18]; future work could compare with the Al potential described in Ref. [19], which has a stacking fault energy of 128 mJ/m2. Appendix A In Fig. 4, the atoms are coloured according to a parameter that is designed to be able to highlight atoms with HCP symmetry even at high temperatures. Similar to the central symmetry parameter described in [26], the atom’s 12 neighbours are put into 6 pairs, each atom paired with the one closest to opposite it. For each pair the deviation from perfect inversion symmetry is measured. Li’s central symmetry parameter (used in Figs. 1 and 2) is the sum of these deviations. An HCP coordinated atom has 3 pairs of atoms in its plane that have inversion symmetry and 3 pairs of atoms from the (1 1 1) planes immediately above and below that do not have inversion symmetry. Taking advantage of this, the parameter used in this paper is the three largest minus the three smallest deviations. This scheme naturally reduces the effect of temperature on the calculation. References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
K. Cheong, E.P. Busso, Acta Mater. 52 (2004) 5665. K. Cheong, E.P. Busso, A. Arsenlis, Int. J. Plasticity 21 (2005) 1797. K. Cheong, E.P. Busso, J. Mech. Phys. Sol. 54 (2006) 671. Y.N. Osetsky, D.J. Bacon, in: H. Kitagawa, Y. Shibutani (Eds.), IUTAM Symposium on Mesoscopic Dynamics in Fracture Process and Strength of Materials, Kluwer, 2004, pp. 193. Y.N. Osetsky, D.J. Bacon, Mater. Sci. Eng. A 400–401 (2005) 353. D.J. Bacon, Y.N. Osetsky, Mater. Sci. Eng. A 400–401 (2005) 374. S. Yip, Nature 391 (1998) 532. M.A. Meyers, A. Mishra, D.J. Benson, Prog. Mater. Sci. 51 (2006) 427. A.S. Khan, Y.S. Suh, X. Chen, L. Takacs, H. Zhang, Int. J. Plasticity 22 (2006) 195. Z.N. Farhat, Y. Ding, D.O. Northwood, A.T. Alpas, Mater. Sci. Eng. A206 (1996) 302. J. Schiøtz, F.D. Di Tolla, K.W. Jacobson, Nature 391 (1998) 561. V. Yamakov, D. Wolf, M. Salazar, S.R. Phillpot, H. Gleiter, Acta Mater. 49 (2001) 2713. V. Yamakov, D. Wolf, S.R. Phillpot, H. Gleiter, Acta Mater. 50 (2002) 5005. V. Yamakov, D. Wolf, S.R. Phillpot, A.K. Mukherjee, H. Gleiter, Nat. Mater. 1 (2002) 1. J. Ray, A. Rahman, J. Chem. Phys. 80 (1984) 4423.
D. Stewart, K.-S. Cheong / Current Applied Physics 8 (2008) 494–497 [16] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, vol. 82, Oxford, 1987. [17] S. Plimpton, J. Comp. Phys. 117 (1995) 1. [18] F. Ercolessi, J. Adams, Europhys. Lett. 26 (1994) 583. [19] X. Liu, F. Ercolessi, J.B. Adams, Model. Simul. Mater. Sci. Eng. 12 (2004) 665. [20] W. Cai, V.V. Bulatov, J. Chang, J. Li, S. Yip, Phys. Rev. Lett. 86 (2001) 5727.
497
[21] W. Cai, V.V. Bulatov, J. Chang, J. Li, S. Yip, Phil. Mag. 83 (2003) 539. [22] W. Cai, V.V. Bulatov, Computer Simulations of Dislocations, Oxford, 2006 (Chapter 5). [23] D. Hull, D.J. Bacon, Introduction to Dislocations, fourth ed., Butterwoth Heinemann, 2001, Chapter 5. [24] J.P. Hirth, J. Lothe, Theory of Dislocations, McGraw-Hill, 1968. [25] T.S. Byun, Acta Mater. 51 (2003) 3063. [26] J. Li, Model. Simul. Mater. Sci. Eng. 11 (2003) 173.
Current Applied Physics 8 (2008) 494–497 www.elsevier.com/locate/cap www.kps.or.kr
Molecular dynamics simulations of dislocations and nanocrystals David Stewart a,*, Ke-Shen Cheong b a
Industrial Research Limited, Gracefield Research Centre, P.O. Box 31-310, Lower Hutt 5040, New Zealand b MPT Solutions Limited, Gracefield Research Centre, P.O. Box 31-310, Lower Hutt 5040, New Zealand Available online 26 October 2007
Abstract Molecular dynamics simulations are used to investigate the mechanical properties of FCC aluminium at the atomic level. As the MD method involves integrating the equations of motion to obtain trajectories for all the particles in the simulation, it is very general. Dislocation–dislocation interactions are simulated to try to find the critical distance at which opposite sign dislocations will mutually annihilate. Simulations of the interaction of a dislocation with a void show that the attraction slows down the dislocation motion, leading to void strengthening. Simulations of the deformation of an aluminium polycrystal with a grain size of 8.6 nm show that not just dislocation motion but also twinning plays a significant role. Twinning is possible in nanocrystalline aluminium because the material can withstand sufficient stress to force partial dislocations separation comparable to the grain size. Ó 2007 Published by Elsevier B.V. PACS: 61.72.Ji; 61.72.Lk; 61.72.Mm; 61.72.Nn; 61.82.Rx; 62.20.x; 62.20.Fe; 82.20.Wt Keywords: Molecular dynamics; Dislocations; Voids; Nanocrystal; Stacking fault; Twinning; Al
1. Introduction Molecular dynamics (MD) is a useful tool for investigating mechanical properties at the nanoscale. The most important deformation mechanism is the motion of dislocations. Dislocation interactions depend on the details of the dislocation core and thus atomistic simulations provide better accuracy than continuum modelling. However, atomistic methods are much more computationally expensive, and thus are limited to smaller regions and shorter timescales. Cheong and Busso’s plasticity model [1–3] predicts the deformation behaviour of polycrystal aggregates by simulating the evolution of densities of edge and screw dislocations in the material. This evolution depends on the distance at which opposite sign dislocations on the same slip system will be able to mutually annihilate. The values for these distances are inferred from temperature-cycling
*
Corresponding author. Tel.: +64 4 931 3715; fax: +64 4 931 3003. E-mail address: [email protected] (D. Stewart).
1567-1739/$ - see front matter Ó 2007 Published by Elsevier B.V. doi:10.1016/j.cap.2007.10.044
experiments. MD simulations offer a way to determine these distances in a more direct way. Irradiated materials contain voids and other obstacles to dislocation glide that can alter the mechanical properties. MD is a good candidate for modelling the dislocation–void interaction, as it depends on the dislocation core structure [4–6]. Nanocrystalline (nc) materials are of interest because they have many properties that are significantly different from the equivalent coarse-grained materials [7,8]. The Hall-Petch relationship states that the strength varies as (grain size)1/2. Nanocrystalline aluminium has been experimentally found to follow this relationship at least down to 15 nm [9,10] and to become more wear resistant with decreasing grain size [10]. The Hall–Petch effect seen in coarse-grained aluminium is a result of dislocation motion. MD simulations on nc Al [13–14] have found deformation by grain-boundary sliding and twinning [13,14], leading to a breakdown of the HP effect. In this paper, we investigate some of the properties of dislocations using MD simulations. First we look at the possible annihilation of opposite dislocations. Then, the
D. Stewart, K.-S. Cheong / Current Applied Physics 8 (2008) 494–497
495
interaction of a dislocation with a void. Then we look at the mechanisms that appear when a nc material is deformed. 2. Methodology
3. Dislocation annihilation We have simulated the interaction of dislocations in aluminium. Two screw dislocations with Burger’s vector 1/2 ½1 1 0 are placed in an FCC crystal of dimensions 35 ½1 1 0 by 30 [1 1 1] by 40 ½1 1 2. The dislocations are in glide planes separated by a number of [1 1 1] planes. 50 MPa of external shear stress is applied to the system. Simulations are done for several separation distances. Fig. 1 shows a simulation where the glide planes are separated by 413 [1 1 1], i.e. 13 atom planes. For screw dislocations at 200 K, annihilation is seen if the number of [1 1 1] planes is 13 or less. If the distance is larger than this, they simply pass by. 4. Dislocation–void interactions We have also simulated the interaction of dislocations with voids in aluminium. Two screw dislocations with Burger’s vector 1/2 ½1 1 0 are placed in an FCC crystal of dimensions 70 ½1 1 0 by 30 [1 1 1] by 20 ½1 1 2. A spherical void with diameter 4 nm is placed in the path of one of the dislocations, with its centre in the glide plane. 50 MPa of external shear stress is applied to the system. The dislocation travels at approximately 0.1 m/s through the crystal. The dislocation bends as it breaks away from the void, as shown in Fig. 2. The dislocation takes an extra 5 ps to get past the void.
Fig. 2. Screw dislocation initially (left) and interacting with a 4 nm void (right) at 20 ps.
5. Nanocrystal deformation A Voronoi construction is used to generate a periodic cell containing 24 grains. For each grain, a random location in the cell is chosen as the grain centre. Then, any point in the cell is considered part of the grain whose centre is closest. The 490,083-atom cell is approximately 20 nm cubed and the grains range in size from 6.8 nm (10,100 atoms) to 10.1 nm (32,708 atoms) with an average of 8.6 nm. Each grain is filled with a randomly orientated FCC lattice. A gap of 0.25 a0 (where a0 is the lattice parameter) is left between grains to avoid generating atoms closer than nearest neighbour distance apart. The cell is equilibrated for a nanosecond of simulation time in order to allow the system to relax into a steady state. The result is a nanocrystal with a range of grain shapes and sizes and a range of different grain boundaries. The cell is then sheared at a constant deformation rate at 200 K. Fig. 3 shows the stress-strain curves for deformations of 5 108 and 5 107 strains per second. As the strain on the nanocrystal is increased, we have observed several different deformation processes. In the initial part of Fig. 3, where the stress is monotonically increasing, dislocations are not obvious; most deformation is by grain boundary sliding as seen in Ref. [11]. In the upper grain shown in Fig. 4, a partial dislocation has nucleated at a grain boundary and crossed the grain, Stress vs Strain 1.4 1.2 1
Stress (GPa)
Our MD simulations use periodic boundary conditions with a Ray–Rahmen flexible simulation cell [15] in order to allow the cell to change size and shape in response to stress. The equations of motion are integrated using a Gear predictor–corrector algorithm [16]. A Nose´-Hoover thermostat is used to control the temperature. The simulations are run in parallel using a domain decomposition algorithm [17]. The Ercolessi-Adams embedded atom potential for aluminium is used [18]. This potential has a stacking fault energy of 104 mJ/m2, compared to experimental measurements of 120–142 mJ/m2 [19].
0.8 0.6 8
Strain rate 5x10 s–1 7 Strain rate 5x10 s–1
0.4 0.2 0 Fig. 1. Periodic cell with two screw dislocations in their initial positions (left) and as they annihilate after 16 ps (right). The glide planes are separated by 13 atomic planes (3.0 nm). Created using atomeye [26].
0
0.02
0.04
0.06
0.08
0.1
Strain Fig. 3. Stress–strain curves for an aluminium Voronoi 8.6 nm nanocrystal deformed at two different strain rates at 200 K.
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D. Stewart, K.-S. Cheong / Current Applied Physics 8 (2008) 494–497
Fig. 4. Voronoi nanocrystal after 8.3% strain. Right image is a close up of the left image. The upper grain shows twinning – the light atoms show the twin planes. The light atoms in the lower grain are a screw dislocation. Created using atomeye [26]. See appendix for details of atom colouring.
creating a stacking fault in its wake. This stacking fault then nucleated twinning. A stacking fault is equivalent to two twinning planes next to each other. These planes move apart via the motion of more partial dislocations, creating a grain between them. The orientation of the new grain is related by mirror symmetry. In the lower grain in Fig. 4, two partial dislocations nucleated and then moved together, creating a full screw dislocation. We have observed cross-slip of screw dislocations that were created this way. 6. Discussion Twinning is not seen in coarse-grained aluminium due to the high stacking fault energy. When a full 1/2 h1 1 0i dislocation dissociates into two 1/6 h1 1 2i partial dislocations, the repulsive force between the partials is balanced by the stacking fault energy, leading to an equilibrium separation distance [23,24]. The Burger’s vectors of the partials differ by 60° and therefore respond to stress differently. This leads to an extra force between the partials proportional to rDb [12,24]. If the resolved shear stress in the glide plane is repulsive and greater than 2c/bp (1.3 GPa for the potential in Ref. [18]), it completely outweighs the attraction; the partials can always lower their energy by moving further apart [12,25]. This level of stress is only possible in the nanocrystalline case, as coarse-grained Al would yield well below this stress level. Even though the choice of mechanism is stress-dependant, both mechanisms were seen at the same level of applied stress. Because the grains are oriented differently, the resolved stress on glide planes in different grains is different. 7. Future work We have simulated dislocation–dislocation and dislocation–void interactions in aluminium. The periodic boundary conditions mean that the dislocation pair is effectively in an infinite lattice of pairs. The elastic interactions between them are long range and lead to
stress variations in the primary cell that alter the result. [20–22]. Ref. [20] shows that for BCC, these spurious effects can be cancelled out if a special aspect ratio is used. More work is needed to clarify the situation for FCC. This will allow the calculation of the dislocation–dislocation annihilation distance and the critical resolved shear stress required for a dislocation to break away from a void-parameters that can be fed into the model of Cheong & Busso. We have simulated the deformation of a nanocrystalline material and observed deformation by twinning and by full dislocations. The effects of grain size and temperature can be investigated. We use the well-established potential from Ref. [18]; future work could compare with the Al potential described in Ref. [19], which has a stacking fault energy of 128 mJ/m2. Appendix A In Fig. 4, the atoms are coloured according to a parameter that is designed to be able to highlight atoms with HCP symmetry even at high temperatures. Similar to the central symmetry parameter described in [26], the atom’s 12 neighbours are put into 6 pairs, each atom paired with the one closest to opposite it. For each pair the deviation from perfect inversion symmetry is measured. Li’s central symmetry parameter (used in Figs. 1 and 2) is the sum of these deviations. An HCP coordinated atom has 3 pairs of atoms in its plane that have inversion symmetry and 3 pairs of atoms from the (1 1 1) planes immediately above and below that do not have inversion symmetry. Taking advantage of this, the parameter used in this paper is the three largest minus the three smallest deviations. This scheme naturally reduces the effect of temperature on the calculation. References [1] [2] [3] [4]
[5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
K. Cheong, E.P. Busso, Acta Mater. 52 (2004) 5665. K. Cheong, E.P. Busso, A. Arsenlis, Int. J. Plasticity 21 (2005) 1797. K. Cheong, E.P. Busso, J. Mech. Phys. Sol. 54 (2006) 671. Y.N. Osetsky, D.J. Bacon, in: H. Kitagawa, Y. Shibutani (Eds.), IUTAM Symposium on Mesoscopic Dynamics in Fracture Process and Strength of Materials, Kluwer, 2004, pp. 193. Y.N. Osetsky, D.J. Bacon, Mater. Sci. Eng. A 400–401 (2005) 353. D.J. Bacon, Y.N. Osetsky, Mater. Sci. Eng. A 400–401 (2005) 374. S. Yip, Nature 391 (1998) 532. M.A. Meyers, A. Mishra, D.J. Benson, Prog. Mater. Sci. 51 (2006) 427. A.S. Khan, Y.S. Suh, X. Chen, L. Takacs, H. Zhang, Int. J. Plasticity 22 (2006) 195. Z.N. Farhat, Y. Ding, D.O. Northwood, A.T. Alpas, Mater. Sci. Eng. A206 (1996) 302. J. Schiøtz, F.D. Di Tolla, K.W. Jacobson, Nature 391 (1998) 561. V. Yamakov, D. Wolf, M. Salazar, S.R. Phillpot, H. Gleiter, Acta Mater. 49 (2001) 2713. V. Yamakov, D. Wolf, S.R. Phillpot, H. Gleiter, Acta Mater. 50 (2002) 5005. V. Yamakov, D. Wolf, S.R. Phillpot, A.K. Mukherjee, H. Gleiter, Nat. Mater. 1 (2002) 1. J. Ray, A. Rahman, J. Chem. Phys. 80 (1984) 4423.
D. Stewart, K.-S. Cheong / Current Applied Physics 8 (2008) 494–497 [16] M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, vol. 82, Oxford, 1987. [17] S. Plimpton, J. Comp. Phys. 117 (1995) 1. [18] F. Ercolessi, J. Adams, Europhys. Lett. 26 (1994) 583. [19] X. Liu, F. Ercolessi, J.B. Adams, Model. Simul. Mater. Sci. Eng. 12 (2004) 665. [20] W. Cai, V.V. Bulatov, J. Chang, J. Li, S. Yip, Phys. Rev. Lett. 86 (2001) 5727.
497
[21] W. Cai, V.V. Bulatov, J. Chang, J. Li, S. Yip, Phil. Mag. 83 (2003) 539. [22] W. Cai, V.V. Bulatov, Computer Simulations of Dislocations, Oxford, 2006 (Chapter 5). [23] D. Hull, D.J. Bacon, Introduction to Dislocations, fourth ed., Butterwoth Heinemann, 2001, Chapter 5. [24] J.P. Hirth, J. Lothe, Theory of Dislocations, McGraw-Hill, 1968. [25] T.S. Byun, Acta Mater. 51 (2003) 3063. [26] J. Li, Model. Simul. Mater. Sci. Eng. 11 (2003) 173.