ARTICLE pubs.acs.org/crystal
An Atomic-Level Mechanism of Annealing Twinning in Copper Observed by Molecular Dynamics Simulation Weiguo Wang,†,‡ Ye Dai,† Jiahao Li,† and Baixin Liu*,† † ‡
Department of Materials Science and Engineering, Tsinghua University, Beijing 100084, China School of Mechanical Engineering, Shandong University of Technology, Zibo 255049, China
bS Supporting Information ABSTRACT: The current experimental methods are not able to reveal the actual processes of atomic movements during twinning and thus are incapable of clarifying the underlying mechanisms of annealing twinning, which are still not clear at present. We developed a method of molecular dynamics simulation to study the mechanism of annealing twinning in copper at an atomic-level. The simulation revealed that a annealing twin can be developed quickly from a pair of grains with Σ3 misorientation interfaced by a (511)/(111) asymmetric boundary. The twinning proceeds by a mechanism in which every three adjacent (511) atomic layers merge into a (111) layer in the (511) side, while the atomic arrangement in the (111) side remains unchanged. Such twinning takes place readily upon annealing at temperatures ranging from 700 to 1300 K, without requiring any extra driving force, indicating that annealing twinning in copper is indeed a thermally activated process with an activation energy estimated to be 0.1 eV. Similar annealing twinning is also observed in another two pairs of grains with Σ3 misorientation interfaced by (822)/(022) and (244)/(200) asymmetric boundaries, respectively, yet their twinning rates are much slower than that of the (511)/(111) grain pair, suggesting a different mechanism governing the process. The simulation also suggested that annealing twinning may involve two separate steps of which one is the formation of grain pairs with Σ3 misorientation and the other is the fine-tuning through which the grain pairs with Σ3 misorientation are converted into ideal annealing twins, which can grow larger with grain growth.
’ INTRODUCTION Twining is not only an essential deformation mode in many kinds of polycrystalline metals with low to medium stacking fault energy (SFE) or with fewer active slipping systems but also of importance in improving the mechanical, electrical, and even chemical properties in materials. For example, Lu et al.13 demonstrated that nanotwinned copper exhibits both higher strength and lower electrical resistivity without decreasing the ductility. Watanabe4 reported that, due to the complete lattice coherency, a twin boundary is almost immune to corrosion attacks. Based on this fact, twin-induced grain boundary engineering (GBE),5,6 which implies enhancing the content of corrosion-resistant twin boundary and its family boundaries through specific thermal-mechanical processing in many low to medium SFE face-centered cubic (fcc) metals so as to improve their performances against intergranular deterioration, is now highlighted in materials science and engineering. Twins can be introduced by both deforming and annealing the materials, which are referred to as deformation and annealing twins, respectively. For fcc metals, the formation mechanism of deformation twins was proposed by Venables,7,8 which is currently accepted as a pole-mechanism.9 It means that the twined atomic layers are formed by a Shockley partial dislocation rotating around a pole of screw dislocation. As to the formation of annealing twins, although a microscopic model was proposed r 2011 American Chemical Society
by Mahajan10 that argued that Shockley partial loops nucleate on consecutive {111} planes by growth accidents occurring on migrating {111} steps associated with a moving grain boundary, it still lacks convincing experimental data. The formation mechanism of annealing twins is still left as an open question. Among the research attempting to understand the formation nature of annealing twinning in the past decades, Randle’s work is the most illuminative. In her study of grain-boundary plane reorientation in copper,11 she found that a great number of Σ3 boundaries developed of which 80% were those incoherent ones (the two adjacent grains or grain pair just have a Σ3 misorientation but do not form a twin) in a sample annealed at high temperature after cold rolling. Her more interesting finding was that most of those incoherent Σ3 boundaries were converted into coherent twin boundaries after subsequent annealing of the sample at a lower temperature. That means most of the grain pairs initially coupled by incoherent Σ3 boundaries were converted into annealing twins. She defined this conversion to be a fine-tuning process but did not give a further explanation at the atomic level. Based on these results, it is believed that although the overall process of annealing twinning may be a complicated Received: January 25, 2011 Revised: March 6, 2011 Published: May 04, 2011 2928
dx.doi.org/10.1021/cg200117x | Cryst. Growth Des. 2011, 11, 2928–2934
Crystal Growth & Design
ARTICLE
Figure 1. Schematic illustrations of the crystallographic geometry of (511)/(111) asymmetric Σ3 relationship (a), MD samples (b), and the identical lattice rotation resulting from the TLMOL occurred in sample A during annealing (c). Although the TLMOL does not involve a rigid lattice rotation actually, its resulting effects are identical to the lattice rotation as described in panel c in which the crystal rotates at first counterclockwise around [011] for 38.94° and then rotates clockwise around [211] for 180°. The [011] direction is perpendicular to the paper plane but out toward the eyes.
one, the so-called fine-tuning starting from a grain pair with an incoherent Σ3 boundary must be a very important mode to form an annealing twin. Obviously, to clarify the underlying atomic movements of the fine-tuning is highly significant for a good understanding overall of the mechanism of annealing twinning. Usually, in situ transmission electron microscopy (TEM) observation is a powerful method to study at the atomic-level the structural evolution of crystalline materials under static or dynamic conditions. However, because the atomic movements are very quick, usually within the time scale of femtoseconds to picoseconds, TEM observations can hardly catch the actual process of atomic movements involved in the fine-tuning. Recently, Janssens et al.12 computed the mobility of flat grain boundaries with molecular dynamics (MD) simulations and obtained some surprising results; Sen and Buehler13,14 carried out MD simulations on deformation of metals in small confinement and showed directly the twinning process via nucleation of partial dislocations. So, it is inspired that the fine-tuning should be simulated by MD method. The most prominent advantage of MD simulation is that it can tailor the movements or real-time positions of each atom at each time step within the time scale of femtoseconds and hence provides a possibility of clarifying the mechanism of the fine-tuning to form an annealing twin.
’ MOLECULAR DYNAMICS SIMULATION Copper was chosen in the present MD simulation. Its interatomic potentials (or many-body potentials) were constructed based on the long-range empirical potentials (LREP),15 which can be expressed analytically as follows: ΦðrÞ ¼ ðr rc1 Þm ðx0 þ x1 r þ x2 r 2 þ x3 r 3 þ x4 r 4 Þ,
0 < r e rc1 ð1Þ
"
# r ΨðrÞ ¼ Rðr rc2 Þ exp β 1 , r0 n
0 < r e rc2
ð2Þ
where Φ(r) and Ψ(r) are pair and electron density terms, rc1 and rc2 are cutoff radii, and x0, x1, x2, x3, x4, r0, R, β, m, and n are potential parameters, which can be determined by fitting the data of physical
Table 1. Detailed Description of the Three MD Samples As Illustrated in Figure 1b (h1k1l1)/
(h2k2l2)/
sample
layers
layers
A
(511)/54
(111)/18
38.3398
39.8438
73.7388
9720
B
(822)/91
(022)/31
30.6717
43.3764
77.5314
8832
C
(122)/91
(100)/31
30.6717
46.0076
109.6459
13248
no. of y0 (Å)
x0 (Å)
z0 (Å)
atoms
properties obtained by both experiments and ab initio calculations.16 Using LREP for MD simulation can effectively avoid the absurdities such as spoiling energy conservation or leading to unphysical behaviors usually encountered by using other short-range potentials if the proper integration methods were not chosen.17,18 The force acting on atom i in a system consisting of N atoms is calculated by Fiforce ¼
N
r
ij ½Φ0 ðrij Þ þ ðFi þ Fj ÞΨ0 ðrij Þ ∑ r ij j6¼ i 0
0
ð3Þ
where rij = ri rj and rij = |rij|. Fi and Fj are functions of Ψ(rij), the detailed description of which can be seen in ref 15. The calculation of the force acting on each atom is the most time-consuming work in the MD simulation. Generally, in formula 3, N is replaced by N0 , which denotes the number of atoms within the range of cutoff radii, and in this way, the calculation time can be reduced drastically. Vectorized link-cell19 method coupled with Gear’s20 predict-corrector was used in the present MD simulations. Three MD samples (A, B, and C) of grain pairs with Σ3 misorientation but, respectively, interfaced by (511)/(111), (822)/(022), and (244)/(200) incoherent asymmetric boundaries ((h1k1l1)/(h2k2l2)), all lying in the [011] zone as shown in Figure 1a,b, were prepared. These three samples are typical for the present MD simulation for understanding the fine-tuning process. During the preparation of the MD samples, the periodic boundary conditions in the three orthogonal directions (o-xyz) were designed carefully, and the detailed description of these samples can be seen in Figure 1b and Table 1. It should be noted that since all three MD samples featured Σ3 misorientation and one-third of the lattice sites of the coupled grains are coincident,21 the interface as depicted in Figure 1b is the first layer of (h2k2l2) in which the number of atoms is three times of that in a (h1k1l1) layer. The lattice constant of copper is a = 3.6147 Å. 2929
dx.doi.org/10.1021/cg200117x |Cryst. Growth Des. 2011, 11, 2928–2934
Crystal Growth & Design
ARTICLE
Figure 3. The evolution of atomic arrangement of the first 18 layers of (511) (Figure 2c)of sample A during annealing at 800 K, showing the process of (511) layers converting into (111) ones: (a) 800 K/2000 time steps; (b) 800 K/10 000 time steps; (c) 800 K/18 000 time steps. One time step = 5 1015 s = 5 fs.
Figure 2. Atomic arrangement of sample A before (a) and after (b) annealing at 700 K for 60 000 time steps (1 time step = 5 1015 s = 5 fs). Panels c and d are the zooming-out of the parts as marked by M and N in panels a and b, respectively. Being different from the work reported by Janssens,12 the artificial or extra driving forces were unnecessary, only the lattice potentials as formulated by eqs 1 and 2 and thermal activation were applied in the present work. This was just the case of the experiment as reported by Randle.11 The present MD simulation was carried out in a wide range of temperatures from 200 to 1300 K. The time-step for MD simulation was chosen to be 5 1015 s (5 fs) based on the criterion by which the total energy of the system remains constant or changes in a acceptable range over the period of time of the total simulation.
’ RESULTS AND DISCUSSION It has been revealed by present MD simulation that the finetuning indeed (to form annealing twins) occurred in the three samples as prepared, but it took place in sample A differently from in sample B and sample C, indicating different mechanisms governing the processes. Figure 2 shows the atomic arrangement of sample A before and after annealing (MD simulation, the same meaning for the following text) at 700 K for 60 000 time steps (Figure 2a,b). It is clear that the (511) layers were converted into (111) ones since the crystalline-plane spacing was changed from (31/2/9)a to (31/2/3)a (Figure 2c,d), while the counterpart (111) layers remained unchanged. After comparison of the atomic arrangement of the (111) layers (as-developed) with that of (111) ones, it is
seen that an ideal annealing twin with an atomic stacking sequence of ABCABC... at both sides of the twinning plane or twin boundary (Figure 2d) was developed. However, the initial interface (or grain boundary) between (511) and (111) did not migrate for any distance during the conversion; it was modified in situ into the (111)/(111) coherent twin boundary. This is a highly interesting phenonmenon behind which the so-called fine-tuning mechanism, which will be clarified in the following, dominated the process. Figure 3 shows the process of (511) layers converting or finetuning into (111) ones at three sequential time nodes when sample A was annealed at 800 K. We can see that the conversion was a time-dependent process that involved a twin nucleation (indicated by an arrow in Figure 3a) at the existing (111) interface (twinning or mirror plane). During annealing, the twin nucleus grew larger and finally converted all the (511) layers into (111) ones (see the ellipse-marked area in Figure 3b,c). So, it is clear that the fine-tuning process of sample A starts from the formation of a twin nucleus. In order to well understand the actual processes of atomic movements to form a twin nucleus so as to clarify the underlying mechanism of the fine-tuning to form an annealing twin in sample A, the real-time positions of the atoms of the smallest unit of the periodic atomic arrangement in the (511) layers must be tailored during MD simulations. Figure 4a gives the atomic arrangements in the xo-y coodinates for the first three (511) layers before annealing (see Figure 2c). It can be seen that the rectangle-marked atomic configuration in Figure 4a is the smallest unit of the periodic atomic arrangement. This unit includes six atoms labeled a, b, c, d, e, and f; they are situated 2930
dx.doi.org/10.1021/cg200117x |Cryst. Growth Des. 2011, 11, 2928–2934
Crystal Growth & Design
ARTICLE
Figure 4. The atomic arrangements in the xo-y coodinates of the first three layers of (511) of sample A before (a) and after (b) annealing at 700 K for 26 000 time steps, 1 time step = 5 1015 s = 5 fs. Atoms af as marked by the rectangle constitute the smallest unit of the periodic atom arrangement. The six atoms of the smallest unit were situated in three (511) layers before annealing as illustrated in Figure 5 in the yo-z coordinate, while after annealing they were aligned in a (111) layer as shown in panel b and Figure 2d.
Figure 5. During annealing atomic movements of the six atoms of the smallest unit as defined in Figure 4a. This schematic Illustration is viewed in the yo-z coodinates. The paper plane is (022), atoms linked by bold-black lines are located in the first (022) plane, which coincides with the paper plane, while those atoms linked by thin-black lines are located in the second (022) plane, which is beneath the paper plane but spaced by (21/2/4)a. Layer 0 of (511) is in coincidence with the first layer of (111) as shown in Figure 2c. The figure shows the atomic movements to convert the (511) layers into (111) ones during annealing.
in the first three (511) layers as shown in Figure 5. After annealing at 700 K for 26 000 time steps, the first three (511) layers have been aligned and merged into a (111) layer as illustrated in Figures 4b and 2d. The new positions of the six atoms can be found inside the marked area in Figure 4b. It is easy to determine that, during the conversion as dipicted in Figures 5 and 6, atoms a and d did not move, atoms b and e took a one-step movement within the (111) plane (note that this (111) plane is defined based on the coordinate of the (511) layers) covering a distance of ΔL1 in the [211] direction, which can be resolved into Δz1 in the [511] and Δy1 in the [255] directions, and atoms c and f took a two-step movement of which the first step was ΔL2 in the [211] direction and the second was Δx in [022] direction. According to the crystallographic geometry as presented by Figure 5, we can find that 2 1 jΔL1 j ¼ jΔy1 þ Δz1 j ¼ ½511a þ ½ 255a 27 54 1 ¼ ½211a ¼ 1:4755 Å 6
Figure 6. During annealing atomic movements of the six atoms of the smallest unit as defined in Figure 4a,b. This figure is viewed in yo-x coordinate: (a) initial position; (b) the atomic movements in yo-x coordinate after movements in z directions as illustrated in Figure 5. Actually the movements in z and y directions are just the resolved components of ΔL1 or ΔL2 proceeding in the (111) plane as shown in Figure 5. (c) Final position.
jΔL2 j ¼ jΔy2 þ Δz2 j ¼
1 ½211a ¼ 0:7378 Å 12
ð5Þ
1 1 jΔxj ¼ ½022a ¼ ½011a ¼ 0:1780 Å 8 4
ð6Þ
¼
ð4Þ 2931
1 1 ½511a þ ½ 255a 27 108
dx.doi.org/10.1021/cg200117x |Cryst. Growth Des. 2011, 11, 2928–2934
Crystal Growth & Design
ARTICLE
Figure 7. Tailoring of real-time positions of the six atoms of the smallest unit as defined in Figure 4a,b during annealing at 700 K. The data sets are recorded every 2000 time steps from starting to 30 000 time steps. One time step = 5 1015 s = 5 fs. The characters af in panel a represent the atoms as shown in Figure 4a,b. (a) Viewed in yo-x coordinate. (b) Viewed in yo-z coordinate.
the absolute distance through which atoms b and e moved was ΔL1 = 1.4755 Å, whereas that for atoms c and f was Δs, which can be determined as follows: jΔsj ¼ jΔL2 þ Δxj ¼
1 1 1 ½211a þ ½011a ¼ ½224a 12 4 12
1 ¼ ½112 ¼ 1:4755 Å 6
ð7Þ
both ΔL1 and Δs are much shorter than the nearest atomic spacing of 2.556 Å in a copper crystal. Summarizing the movements of the six atoms of the smallest unit during annealing while examining the morphology of the twin nucleus as shown in Figure 3a shows that the annealing twin nucleus was formed by the side of the pre-existing (111) interface through the atomic movement and layer mergence as described by both the eqs 47 and the Figures 46. Once its dimension exceeded the critical size, which can be determined through further computations but not to be discussed in the present paper, the twin nucleus would grow up quickly and convert the (511) layers into (111) ones through the mechanism of every three adjacent (511) layers merging into a (111) one in o-z direction (Figure 5). This can be confirmed by examining the atomic movement of other (511) layers above the first three such as the layers numbered 46, 79, and so on in Figure 2c. For convenience, such conversion of finetuning in sample A is tentatively termed as TLMOL (three layers merged into one layer). Figure 7 shows the tailoring of real-time positions of the six atoms of the smallest unit as defined in Figure 4a, b during annealing at 700 K; it agrees well with the analyses above. Since it takes place readily in the temperature ranging from 700 to 1300 K without requiring any extra driving force, TLMOL is of course a thermally activated process of which the activation energy E can be preliminarily determined to be 0.1 eV by fitting the results of MD simulation based on the following equation: E ð8Þ υ ¼ μ exp KT
where υ is the conversion rate, μ is a constant, K is Boltzmann constant, and T is the temperature. We can see that the activation energy is much lower than that of self-diffusion of copper, which was reported to be 2.4 eV elsewhere,22 indicating that TLMOL occurs more easily than self-diffusion does in copper. It agrees well with the results as determined in the foregoing text that the distances of atomic movement in TLMOL are much shorter than the nearest atomic spacing, which is the shortest distance of movement for the self-diffusion. Obviously, TLMOL is identical to the lattice rotation as described in Figure 1c though it does not involve a rigid rotation in actuality. When comparing the crystallographic geometry before and after TLMOL, we can find that TLMOL also took place in the (255) layers in o-y direction (Figures 1b and 5) by which every adjacent three (255) layers merged into a (211) layer, while the (022) layers in the o-x direction remained unchanged. It is easy to know that TLMOL does not render any net strain in the three orthogonal directions (o-xyz); this must be one more account for its easy activation. In fact, the two vectors ΔL1 and Δs (see eqs 4 and 7) that denote the atomic movements in TLMOL are the Burger’s vectors of Schockley23 partial dislocations (Figure 4a), which pre-exist at the partially matched (511)/(111) interface. These partial dislocations are intrinsically imparted at the specific (511)/(111) incoherent Σ3 interface (boundary) according to the fundamentals of coincidence site lattice (CSL).21 The atomic movement involved in TLMOL is actually the movement of the partial dislocations, which is very easy to activate. Figure 8 shows the atomic arrangement of the central part layers of samples B and C after annealing; it can be seen that annealing twinning also occurred in both the samples but was quite different from that in sample A. On one hand, the onset of annealing twinning incubated for longer time and occurred at higher temperature; for example, no any change of atomic arrangement was observed in sample C even though it had been annealed at 900 K for 60 000 time steps, whereas the twinning was finished in sample A only after 26 000 time steps annealing at 700 K. On the other hand, the twin boundaries spontaneously located at the pre-existing {111} planes (Figure 8a,b) and deviated from the position of initial interfaces for 35.26° and 54.74°, respectively, in samples B and C. These results indicate that the annealing twinning is more difficult in samples B and C, and it must be dominated by other mechanisms, which need further clarification. Nevertheless, the annealing twinning occurring in samples B and C is also a process of fine-tuning but differs from TLMOL; it may involve the movement of more atoms and longer distances, and the mechanism underlying it may be elucidated by a more explicit MD simulation, which will not be discussed in the present paper. So, it should be highlighted that the TLMOL as we observed in the present work is a highly interesting finding although it only occurred in the specific grain pair with a (511)/(111) incoherent Σ3 boundary. This new mechanism of annealing twinning as we revealed at the atomic level suggests that an annealing twin can be easily and quickly formed through the fine-tuning (TLMOL) in fcc polycrystalline materials when any two grains contact each other with (511)/(111) incoherent Σ3 boundary during deformation plus annealing. It is most likely that the nearly 20% of twin boundaries out of the total Σ3, as Randle11 obtained by annealing a cold-rolled pure copper sample at 1173 K for a short term, were achieved mostly through TLMOL. This is a most probable explanation for the fact that Randle11 could not observe any (511)/(111) incoherent Σ3 boundaries in that sample either 2932
dx.doi.org/10.1021/cg200117x |Cryst. Growth Des. 2011, 11, 2928–2934
Crystal Growth & Design
ARTICLE
Figure 8. The atomic arrangement of samples B (a) and C (b) after annealing at 1000 K for 60 000 and 1200 K for 14 000 time steps, respectively. The positions marked with red dots are twin boundaries. The atoms located inside the ellipse zone are at the transition state showing the morphology of some diffusion.
before or after subsequent annealing at lower temperature. So, the next step is to further study the conditions for the formation of grain pairs with (511)/(111) incoherent Σ3 boundaries, it will be highly significant for twinning control, as well as for the GBE research, which is expected to better the performance of the materials to a surprising grade.13,5,6
’ CONCLUDING REMARKS As has been mentioned, the conventional opinion10 proposed that the annealing twins are produced by migrating random boundaries through fault stacking on the {111} steps. It stresses that boundary migration is the prerequisite and seems reasonable because annealing twins must be introduced in the process of annealing, which indeed involves boundary migration. However, Randle’s recent work24 with a pure nickel sample demonstrated that during a steady-state grain growth from an average grain size of 70 to 150 μm, there were no more annealing twins developed, and annealing twins increased sharply only in a very narrow range of grain sizes (6470 μm) below the temperature in which fast grain growth took place. This suggests the annealing twinning could have happened almost independently of boundary migration. When we consider the fact that annealing twins possess long and straight morphology and the twin boundary does not constitute a part of the boundary networks, when we consider the results of twinning behaviors in a copper tricrystal reported recently by Miura,25 and when we are reminded of the finetuning as observed in the present work, we may think that the triple junctions in polycrystalline metals play a very important role in annealing twinning. In other words, annealing twinning may involve two separate steps. One is the formation of grain pairs with Σ3 misorientation through the lattice rotation at the strain-concentrated (or correctly deformed) triple junctions, as many as 16 selections of Σ3 misorientation26 in fcc metals provide the possibilities for this step. The other is the fine-tuning through which the grain pairs with Σ3 misorientation are converted into ideal annealing twins, which can grow larger with grain growth. Of course, the easiness of fine-tuning differs from one to another for the 16 selections of Σ3 misorientation, just like the three cases encountered in the present work. When the finetuning is very easy (like TLMOL in present work), there will be no observable intervals between the two separate steps and the annealing twins may be formed simultaneously with the formation of grain pair of Σ3 misorientation; otherwise, the grain pairs with incoherent Σ3 boundaries between them will be left,27 and
subsequent annealing is necessary for converting them into annealing twins. Of course, considering the fact that small prestrain usually enhances the occurrence of twinning in copper during annealing,28 the ABC method,29 which involves the application of strains in the system and can reach longer time scale in MD simulation, may be employed. It will probably enable us to understand in a comprehensive way the overall processes of the formation of annealing twins.
’ ASSOCIATED CONTENT
bS
Supporting Information. Atomic arrangements before and after annealing at 700 K were viewed on the yo-z plane. This material is available free of charge via the Internet at http:// pubs.acs.org.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT The authors are grateful for the financial support from the National Natural Science Foundation of China (Grants 50771060, 50871058, and 50971072), the Ministry of Science and Technology of China (Grant 2006CB605201), the Ministry of Education of China (Grant 200800030054), and the Administration of Tsinghua University. W. G. Wang is also much grateful to the Shandong Provincial Government for the financial support through the program of domestic visiting scholarship. ’ REFERENCES (1) Lu, K.; Lu, L.; Suresh, S. Science 2009, 324, 349–352. (2) Lu, L.; Shen, Y. F.; Chen, X. H.; Qian, L. H.; Lu, K. Science 2004, 304, 422–426. (3) Li, X. Y.; Wei, Y. J.; Lu, L.; Lu, K.; Gao, H. J. Nature 2010, 464, 877–880. (4) Watanabe, T. Res Mech. 1984, 11, 47–84. (5) Randle, V. Acta Mater. 2004, 52, 4067–4081. (6) Shimada, M.; Kokawa, H.; Wang, Z. J.; Sato, Y. S.; Karibe, I. Acta Mater. 2002, 50, 2331–2341. (7) Venables, J. A. Philos. Mag. 1961, 6, 379–396. (8) Venables, J. A. J. Phys. Chem. Solids 1964, 25, 693–700. 2933
dx.doi.org/10.1021/cg200117x |Cryst. Growth Des. 2011, 11, 2928–2934
Crystal Growth & Design
ARTICLE
(9) Huang, C. X.; Wang, K.; Wu, S. D.; Zhang, Z. F.; Li, G. Y.; Li, S. X. Acta Mater. 2006, 54, 655–665. (10) Mahajan, S.; Pande, C. S.; Imam, M. A.; Rath, B. B. Acta Mater. 1997, 45, 2633–2638. (11) Randle, V.; Davies, P.; Hulm, B. Philos. Mag. A 1999, 79, 305–316. (12) Janssens, K. F.; Olmsted, D.; Holm, E.; Foiles, S.; Plimpton, S.; Derlet, A. Nat. Mater. 2006, 5, 124–127. (13) Sen, D.; Buehler, M. J. Phys. Rev. B 2008, 77, No. 195439. (14) Buehler, M. J. Atomistic Modeling of Materials Failure; Springer: New York, 2008. (15) Li, J. H.; Dai, X. D.; Liang, S. H.; Tai, K. P.; Kong, Y.; Liu, B. X. Phys. Rep. 2008, 455, 1–134. (16) Dai, Y.; Li, J. H.; Che, X. L.; Liu, B. X. J. Phys. Chem. B 2009, 113, 7282–7290. (17) Johnson, R. A.; Oh, D. J. J. Mater. Res. 1989, 4, 1195–1201. (18) Dai, X. D.; Kong, Y.; Li, J. H.; Liu, B. X. J. Phys.: Condens. Matter. 2006, 18, 4527–4542. (19) Fincham, D.; Heyes, D. M. Dynamical Processes In Condensed Matter; Evans, M. W., Ed.; Advances in Chemical Physics, Vol LXIII; Wiley: New York, 1985. (20) Berendsen, H. J. C.; Van Gunsteren, W. F. Practical Algorithms For Dynamic Simulations; Proceedings of the International School of Physics “Enrico Fermi”, North-Holland: Amsterdam, 1986. (21) Brandon, D. G. Acta Met. 1966, 14, 1479–1484. (22) Shockley, W.; Nix, F. C. Phys. Rev. 1939, 56, 13–21. (23) Song, S. G.; Cole, J. I.; Bruemmer, S. M. Acta Mater. 1997, 45, 501–511. (24) Randle, V.; Rios, P. R.; Hu, Y. Scr. Mater. 2008, 58, 130–133. (25) Miura, H.; Sakai, T.; Jonas, J. J. Scr. Mater. 2006, 55, 167–170. (26) Wolf, U.; Ernst, F.; Muschik, T.; Finnis, M. W.; Fischmeister, H. F. Philos. Mag. A 1992, 66, 991–1016. (27) Wang, W. G.; Zhou, B. X.; Rohrer, G. S.; Guo, H.; Cai, Z. X. Mater. Sci. Eng. A 2010, 527, 3695–3706. (28) Cai, Z. X.; Wang, W. G.; Fang, X. Y.; Guo, H. Acta Metall. Sin. 2010, 46, 769–774. (29) Lau, T. T.; Kushima, A.; Yip, S. Phys. Rev. Lett. 2010, 104, No. 175501.
2934
dx.doi.org/10.1021/cg200117x |Cryst. Growth Des. 2011, 11, 2928–2934
