- Email: [email protected]
Atomic dynamics of grain boundaries in bulk nanocrystalline aluminium: A molecular dynamics simulation study
Computational Materials Science 108 (2015) 177–182
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Atomic dynamics of grain boundaries in bulk nanocrystalline aluminium: A molecular dynamics simulation study Z.Y. Hou a,⇑, Z.A. Tian b, Y.F. Mo b, R.S. Liu b, J.G. Wang a, X.M. Shuai a, K.J. Dong c a
Department of Applied Physics, Chang’an University, Xi’an 710064, China School of Physics and Microelectronics Science, Hunan University, Changsha 410082, China c Institute for Infrastructure Engineering, University of Western Sydney, Penrith, NSW 2751, Australia b
a r t i c l e
i n f o
Article history: Received 9 March 2015 Received in revised form 12 June 2015 Accepted 26 June 2015
Keywords: Atomic dynamics Grain boundaries Bulk nanocrystalline aluminium Molecular dynamics simulation
a b s t r a c t Dynamics of grain boundary (GB) atoms in the bulk nanocrystalline aluminium is investigated by means of a large-scale molecular dynamics simulation. It is found that the GB atoms in the nanocrystalline aluminium display the glassy dynamics. Their dynamic features are, on one hand, similar to those in the glass systems and bicrystal GBs, in terms of the time-correlation functions of mean-square displacement (MSD) and non-Gaussian parameter (NGP). But on the other hand, these GB atoms also demonstrate some different dynamic behaviors due to the more complicated microstructure. In particular, the immobile GB atoms are localized into their equilibrium positions due to the strong cage effect, and they are mainly at the surfaces of grains, especially the large grains, while the mobile GB atoms are active and can easily hop away from the cage around them. These mobile GB atoms gather together at the surfaces of small grains and the triple junction regions of GBs, and they form some abnormally big clusters. The size distribution of these clusters significantly deviates from the power law. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Bulk nanocrystalline materials have been a subject of extensive research in recent decades, due to their unique physical and chemical properties [1]. This class of materials is distinguished by a significant volume of grain boundaries (GBs). The dynamic behaviors of GB atoms, such as diffusion, migration, and sliding, play a pivotal role in the deformation and grain growth of bulk nanocrystalline metals [2–5]. Many studies [3–8] have been performed to elucidate the spatiotemporal evolution of GBs in the bulk nanocrystalline metals, but our understanding of GB dynamics at atomic level is still limited owing to the disparities in the length and time scales associated with its experimental observation [9]. The width of GBs in the bulk nanocrystalline structure usually is about 2–4 atomic layers, so it is difficult to experimentally probe its microstructure and dynamic evolution at the atom level [10]. Molecular dynamics (MD) simulation is an effective tool to study the microstructure and dynamics of bulk nanocrystalline materials, as a very useful extension of experimental and analytical investigations. Some simulation studies [11–13] showed that the microstructure of GBs in bulk nanocrystalline metals is disordered, quite similar to the glass systems. Other simulations [14–16] also ⇑ Corresponding author. E-mail address: [email protected] (Z.Y. Hou). http://dx.doi.org/10.1016/j.commatsci.2015.06.039 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
suggested the similarities between the bicrystal GBs and the glass systems in dynamics, evidenced by the dynamic heterogeneous and the string-like cooperative atomic motion. And the glassy dynamics of GBs has been experimentally confirmed in some colloidal systems [17,18]. However, previous dynamic studies of GBs focused on the bicrystal GBs which comprise only two grains and their interfacial regions, while the atomic dynamics of GBs in the bulk nanocrystalline metals was seldom reported. The microstructure and dynamics of GBs in the realistic bulk nanocrystalline metals are more complex than those in the bicrystal GBs. For example, the GBs in the bulk nanocrystalline structure contain lots of triple junction (TJ) regions which usually have particular kinetic and thermodynamic properties [19–22]. As the nanograins in the bulk nanocrystalline structure have different sizes and random crystallographic orientation, the GB dynamics were found to vary with the misorientation angle between adjacent grains [17]. Furthermore, recent studies [23–25] demonstrated that a large percentage of GB atoms are in the regions with significant crystalline order, while the disordered amorphous-like atoms mainly locate at the TJ regions. In our previous work [25,26], the bulk nanocrystalline aluminium was produced by means of a large-scale MD simulation of the rapid solidification process of liquid, and the microstructure of GBs is investigated in detail. It was found that the
178
Z.Y. Hou et al. / Computational Materials Science 108 (2015) 177–182
microstructure features of grains and GBs obtained in our simulations are consistent with those produced in some experiments and geometrically constructed by the Voronoi cell method. In this work, we further investigate the atomic dynamics of these GBs. It is found that the GB atoms display the dynamics similar to those in the glass systems and bicrystal GBs in some aspects, but also exhibit some different dynamic features from the latter. 2. Computational methods 2.1. Molecular dynamics simulation The rapid solidification process of liquid aluminium is simulated using the LAMMPS codes [27] for a system containing 1,048,576 atoms in a cubic box with periodic boundary conditions. The interaction potential adopted here is the embedded atoms method (EAM) potential proposed by Mendelev et al. [28]. As shown in Refs. [25,26], the bulk nanocrystalline aluminium is obtained below the temperature 473 K, when the liquid Al is quenched from 1273 K at a cooling rate of 1 1012 K/s. The local atomic clusters in the system are identified in terms of a recently developed method of largest standard cluster analysis (LSCA) [29], in which the neighbor of an atom are identified with a parameter-free topological criterion rather than a fixed cut-off distance rc. The method of identified grain and GB atoms are described in Refs. [25,26]. In the nanocrystalline structure, as shown in Refs. [25,26], the nanograins with an average size of 6.4 nm are separated by high-angle GBs. The GB regions neighboring the grains display FCC-like short-to-long range order. The TJ regions of GB are not disordered like liquid, but present icosahedral- and BCC-like short-range order. To investigate the dynamic properties of GB atoms, the nanocrystalline structures at different temperatures, every 100 K from 473 K to 273 K, are respectively relaxed 500 ps under an isothermal–isochoric (NVT) ensemble. More detailed information on the simulation methods can be obtained in Refs. [25,26]. 2.2. Characterization of dynamics The mean-square displacement (MSD) [30] is usually used to study the dynamic motion of atoms, which is defined as
hr2 ðtÞi ¼
N D E 1X jri ðtÞ ri ð0Þj2 ; N i¼1
ð1Þ
Fig. 1. Time dependence of MSD for the GB atoms in the bulk nanocrystalline aluminium at different temperatures.
the short time from the beginning (t < 0.1 ps), all curves show a power-law behavior which indicates the ballistic motion of atoms. At the intermediate time, the atoms are trapped in the transient cages formed by their neighbor atoms, and the trapped atoms need some time to escape from the cages. So a plateau appears on the MSD curve due to the cage effect, and it becomes pronounced with the decrease of temperature. At the final time, the curves increase with the slope n < 1, indicating the onset of cage-breaking rearrangement. The time window correlated with the cage effect is usually called the b-relaxation regime and the following time window is the a-relaxation regime [30]. To further assess the dynamics of GB atoms in the bulk nanocrystalline aluminium, its dynamic heterogeneity is investigated by means of the NGP. As shown in Fig. 2, when t < 0.1 ps, all NGPs at different temperatures are close to zero, indicating the Gaussian distribution of the vibration displacements of GB atoms. During the following b-relaxation window, all NGP curves rise monotonically to a maximum value, and the position of maximum shifts toward the longer time as the temperature decreases. This suggests that the GB atoms in the bulk nanocrystalline aluminium exhibit dynamic heterogeneity in the b-relaxation window, and such heterogeneity increases with the enhancement of cage effect.
where N is the number of atoms in the system, ri(t) is the position vectors of the ith atom at time t. The non-Gaussian parameter (NGP) [31] is frequently used to quantify the dynamic heterogeneity of supercooled liquids, and is defined as:
a2 ðtÞ ¼
3hr 4 ðtÞi 5hr 2 ðtÞi2
1;
ð2Þ
where hr4(t)i is the mean quartic displacement. The heterogeneity of the atom motion can thus be inferred from the time evolution of a2, which is zero for a Gaussian distribution, and a2 > 0 for non-Gaussian behavior. The a2 reaches a maximum for a characteristic time named t*. 3. Results and discussion 3.1. Dynamics of grain boundaries Fig. 1 shows the time dependence of MSD for the GB atoms in the bulk nanocrystalline aluminium at different temperatures. In
Fig. 2. Time dependence of non-Gaussian parameter (NGP) for GB atoms in the bulk nanocrystalline aluminium at different temperatures.
Z.Y. Hou et al. / Computational Materials Science 108 (2015) 177–182
179
Fig. 3. Probability distribution of displacements of GB atoms P(Dx) over Dt = t⁄ in the bulk nanocrystalline aluminium at 273 K. The solid line is the Gaussian fit. Particles outside the shaded region are the top 5% most mobile GB atoms.
Fig. 6. Time dependence of MSD for the mobile and immobile GB atoms, all GB atoms and grain atoms in the bulk nanocrystalline aluminium at 273 K.
Fig. 4. Distributions of mobile and immobile GB atoms around the grains with different sizes.
Fig. 7. Probability distribution P(n) of cluster size n for the mobile GB atoms in the nanocrystalline aluminium at 473 K and 273 K.
The MSD and NGP of GBs in the bulk nanocrystalline aluminium are similar to those characterized in the metal glass aluminium [32]. This demonstrates the glassy dynamics of GBs in the nanocrystalline structure. However, its b-relaxation regime is more distinct than those in the colloidal polycrystals [17,18]. This may be caused by the strong cohesive interactions existing in the metal systems which would result in slow dynamics [33]. 3.2. String-like cooperative motion
Fig. 5. A cross section of the nanocrystalline aluminium (at 273 K) through its center. The grain atoms are presented in gray dots; the GB atoms are presented in gray spheres, in which the mobile and immobile GB atoms are respectively displayed in cyan and purple colours. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Cooperative motion is one of the most characteristic features of the dynamics of supercooled liquids and glasses [32–36]. Particularly, the string-like cooperative motion of mobile atoms has been observed in the atomic dynamics of supercooled liquids and glasses [34–36], GB self-diffusion [37], and interfacial dynamics of nanoparticles [38]. These cooperative motion regions usually exhibit a power size distribution, and their average extent grows upon cooling [34–37]. To investigate the cooperative dynamics of GB atoms in the bulk nanocrystalline aluminium, the mobile atoms in the GBs are identified firstly. Fig. 3 shows the probability distribution of displacements of GB atoms P(Dx) over Dt = t⁄ at 273 K. It can be found that 95% GB atoms display diffusive dynamics, while the top 5% mobile GB atoms show non-Gaussian behavior. The proportion of GB atoms with non-Gaussian behavior is smaller than that observed in the
180
Z.Y. Hou et al. / Computational Materials Science 108 (2015) 177–182
Fig. 8. String-like cooperative motion of clusters combined of mobile GB atoms in the bulk nanocrystalline aluminium at 273 K. The atom radii in the three clusters are different for better visualization. The atoms in each clusters at t = 0 and t⁄ are respectively displayed in pink and cyan colours. The arrows in the clusters indicate the displacements of cluster atoms between t = 0 and t⁄, and they are plotted being based on the magnitudes and directions of their displacements. (a) Schematic configuration of a typical cluster. (b) Square displacements of the cluster atoms in (a) during the relaxation process. (c) Schematic configuration of another typical cluster with two ‘‘strings’’ indicated by the arrows with different colours. (d) Snapshot of the biggest cluster in the system. It is combined by many different ‘‘strings’’. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Z.Y. Hou et al. / Computational Materials Science 108 (2015) 177–182
colloidal polycrystal (10%) [17], but is the same as the supercooled liquids and glasses [32–36]. So we select the 5% GB atoms with the largest displacements as the mobile atoms, and the 5% GB atoms with the smallest displacements as the immobile atoms, just as the method proposed by Donati [34]. Two mobile atoms are considered to belong to a same cluster if they are neighbors. For the mobile GB atoms, 55.1% of them are located at the nearest-neighbor coordination shell around grains, and 35.2% are at the TJ GB regions. Furthermore, the proportion of mobile GB atoms around the small grains is notably higher than that around the large grains (see Fig. 4). For the immobile GB atoms, 84.9% of them are located at the nearest-neighbor coordination shell around grains, while only 6.4% are at the TJ GB regions. And the proportion of immobile GB atoms around the large grains is slightly higher than that around the small grains. From the cross section of the nanocrystalline structure (at 273 K) through its center, as shown in Fig. 5, it can also be found that the mobile GB atoms are mainly located at the surfaces of small grains and the TJ GB regions, while the immobile GB atoms are mainly at the surfaces of grains, especially the large grains. This results are consistent with the facts that the GB atoms at the surfaces of small grains and the TJ GB regions diffuse faster than others [19–22,39,40]. Fig. 6 further shows the differences of dynamic behaviors of different GB atoms. It can be found that the plateau in the MSD curve for the immobile atoms is close to a horizontal line and the cage-breaking rearrangement are not occurred, which is similar to that for the grain atoms. While the plateau in the MSD curve for the mobile atoms is less obvious due to the weak cage effect. This indicates that the immobile GB atoms are localized into their equilibrium positions due to the strong cage effect, while the mobile GB atoms are active and can easily hop away from the cage around them. Fig. 7 shows the size distribution of the clusters combined by the mobile GB atoms. It can be found that the average size of clusters increases with decreasing temperature, which is consistent with that in the supercooled liquids and glass systems [34–36]. It can also be found that most of these clusters have a small size (n < 40), and their probability distribution P(n) decreases with n in an power law as also observed in the supercooled liquids and glasses. But P(n) for the big clusters (n > 40) significantly deviates from the power law, the sizes of these clusters are abnormally larger than others comparing with that in the glass systems and bicrystal GBs. This means that most mobile GB atoms in the nanocrystalline aluminium gather together and form some big clusters (see Fig. 5). Fig. 8(a) shows the schematic configuration of an arc-shaped string-like cluster combined by the mobile GB atoms. All cluster atoms vibrate in small amplitudes during the initial relaxation stage (see Fig. 8(b)). Then their square displacements rapidly increase at about the same time t 20 ps. This indicates that all cluster atoms hop together away from their original positions at t 20 ps, and then diffuse along the string indicated by the arrows in Fig. 8(a), showing a cooperative motion in a string-like manner. Similarly, the cluster in Fig. 8(c) also displays a string-like cooperative motion of atoms, while it is composed of two different ‘‘strings’’ indicating by the arrows with different colours. The biggest cluster found in the system is shown in Fig. 8(d), which contains a lot of ‘‘strings’’ but also exhibits a string-like cooperative motion.
4. Conclusions To investigate the atomic dynamics of GBs in the bulk nanocrystalline structure, the bulk nanocrystalline aluminium was produced by molecular dynamics simulation of the solidification
181
process of a large-scale system, and the dynamic properties of GB atoms have been characterized and investigated by various methods. It is found that the time-correlation functions of MSD and NGP for the GB atoms in the bulk nanocrystalline aluminium display the similar features as those in the metal glass aluminium and bicrystal GBs. In particular, the MSD curve shows a pronounced plateau at the intermediate time due to the cage effect; the NGP curve rise monotonically to a maximum value in the b-relaxation window; and the mobile GB atoms undergo the cooperative motion in a string-like manner. These dynamic features become more pronounced with decreasing temperature. On the other hand, the GB atoms in the nanocrystalline structure have some different dynamic features comparing with those in the glass systems and bicrystal GBs. In particular, the atoms in the different regions of GB display notably different dynamic behaviors. The immobile GB atoms are localized into their equilibrium positions due to the strong cage effect, and they are mainly at the surfaces of grains, especially the large grains, while the mobile GB atoms are active and can easily hop away from the cage around them. The mobile GB atoms in the bulk nanocrystalline structure gather together at the surfaces of small grains and the TJ GB regions, and form some abnormally big clusters comparing with those in the glass systems and bicrystal GBs. The size distribution of the mobile GB clusters satisfies a power law for the small clusters with size n < 40, but significantly deviates from the power law for large clusters. Acknowledgments The authors would like to acknowledge the support provided by the National Natural Science Foundation of China (Grant Nos. 51101022 and 51071065). This work is also supported by the Fundamental Research Funds for the Central Universities of China (Grant Nos. 310812152001 and 2013G1121082). References [1] C.C. Koch, J. Mater. Sci. 42 (2007) 1403. [2] B.N. Kim, K. Hiraga, Y. Sakka, B.W. Ahn, Acta Mater. 47 (1999) 3433. [3] A.J. Haslam, V. Yamakov, D. Moldovan, D. Wolf, S.R. Phillpot, H. Gleiter, Acta Mater. 52 (2004) 1971. [4] Z. Shan, E.A. Stach, J.M.K. Wiezorek, J.A. Knapp, D.M. Dollstaedt, S.X. Mao, Science 305 (2004) 654. [5] S.V. Bobylev, N.F. Morozov, I.A. Ovid’Ko, Phys. Rev. Lett. 105 (2010) 0555504. [6] V. Yamakov, D. Moldovan, K. Rastogi, D. Wolf, Acta Mater. 54 (2006) 4053. [7] D. Jang, M. Atzmon, J. Appl. Phys. 99 (2006) 083504. [8] Y.T. Zhu, X.Z. Liao, X.L. Wu, Prog. Mater Sci. 57 (2012) 1. [9] S. Gokhale, K.H. Nagamanasa, R. Ganapathy, A.K. Sood, Soft Matter 9 (2013) 6634. [10] H. Van Swygenhoven, Science 296 (2002) 66. [11] S.R. Phillpot, D. Wolf, H. Gleiter, J. Appl. Phys. 78 (1995) 847. [12] P. Keblinski, S.R. Phillpot, D. Wolf, H. Gleiter, Phys. Rev. Lett. 77 (1996) 2965. [13] J. Löffler, J. Weissmüller, Phys. Rev. B 52 (1995) 7076. [14] H. Zhang, D.J. Srolovitz, J.F. Douglas, J.A. Warren, Proc. Natl. Acad. Sci. USA 106 (2009) 7735. [15] H. Zhang, D.J. Srolovitz, J.F. Douglas, J.A. Warren, Phys. Rev. B 74 (2006) 115404. [16] H. Zhang, M.I. Mendelev, D.J. Srolovitz, Acta Mater. 52 (2004) 2569. [17] K.H. Nagamanasa, S. Gokhale, R. Ganapathy, A.K. Sood, Proc. Natl. Acad. Sci. USA 108 (2011) 11323. [18] T.O.E. Skinner, D.G.A.L. Aarts, R.P.A. Dullens, J. Chem. Phys. 135 (2011) 124711. [19] M.R. Chellali, Z. Balogh, H. Bouchikhaoui, R. Schlesiger, P. Stender, L. Zheng, G. Schmitz, Nano Lett. 12 (2012) 3448. [20] T. Frolov, Y. Mishin, Phys. Rev. B 79 (2009) 174110. [21] M.R. Chellali, Z. Balogh, L. Zheng, G. Schmitz, Scripta Mater. 65 (2011) 343. [22] O.K. Johnson, C.A. Schuh, Acta Mater. 61 (2013) 2863. [23] H. Van Swygenhoven, D. Farkas, A. Caro, Phys. Rev. B 62 (2000) 831. [24] Z. Wu, Y. Zhou, X. Zhang, S. Wei, D. Chen, Appl. Phys. Lett. 84 (2004) 4442. [25] Z. Hou, Z. Tian, Y. Mo, R. Liu, Comput. Mater. Sci. 92 (2014) 199. [26] Z. Hou, Z. Tian, R. Liu, K. Dong, A. Yu, Comput. Mater. Sci. 99 (2015) 99. [27] S.J. Plimpton, J. Comp. Phys. 117 (1995) 1. [28] M.I. Mendelev, M.J. Kramer, C.A. Becker, M. Asta, Phil. Mag. 88 (2008) 1723. [29] Z.A. Tian, R.S. Liu, K.J. Dong, A.B. Yu, EPL 96 (2011) 36001. [30] W. Kob, J. Phys.: Condens. Matter 11 (1999) R85.
182 [31] [32] [33] [34]
Z.Y. Hou et al. / Computational Materials Science 108 (2015) 177–182
A. Rahman, Phys. Rev. 136 (1964) A405. M. Li, C.Z. Wang, M.I. Medelev, K.M. Ho, Phys. Rev. B 77 (2008) 184202. Z. Zhang, P.J. Yunker, P. Habdas, A.G. Yodh, Phys. Rev. Lett. 107 (2011) 208303. C. Donati, S.C. Glotzer, P.H. Poole, W. Kob, S.J. Plimpton, Phys. Rev. E 60 (1999) 3107. [35] E.R. Weeks, J.C. Crocker, A.C. Levitt, A. Schofield, D.A. Weitz, Science 287 (2000) 627.
[36] [37] [38] [39] [40]
Y. Gebremichael, M. Vogel, S.C. Glotzer, J. Chem. Phys. 120 (2004) 4415. Y. Mishin, Defect Diffus. Forum 363 (2015) 1. H. Zhang, P. Kalvapalle, J.F. Douglas, Soft Matter 6 (2010) 5944. B. Arab, S. Sanjabi, A. shokuhfar, Mater. Lett. 65 (2011) 712. D. Prokoshkina, V.A. Esin, G. Wilde, S.V. Divinski, Acta Mater. 61 (2013) 5188.
Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Atomic dynamics of grain boundaries in bulk nanocrystalline aluminium: A molecular dynamics simulation study Z.Y. Hou a,⇑, Z.A. Tian b, Y.F. Mo b, R.S. Liu b, J.G. Wang a, X.M. Shuai a, K.J. Dong c a
Department of Applied Physics, Chang’an University, Xi’an 710064, China School of Physics and Microelectronics Science, Hunan University, Changsha 410082, China c Institute for Infrastructure Engineering, University of Western Sydney, Penrith, NSW 2751, Australia b
a r t i c l e
i n f o
Article history: Received 9 March 2015 Received in revised form 12 June 2015 Accepted 26 June 2015
Keywords: Atomic dynamics Grain boundaries Bulk nanocrystalline aluminium Molecular dynamics simulation
a b s t r a c t Dynamics of grain boundary (GB) atoms in the bulk nanocrystalline aluminium is investigated by means of a large-scale molecular dynamics simulation. It is found that the GB atoms in the nanocrystalline aluminium display the glassy dynamics. Their dynamic features are, on one hand, similar to those in the glass systems and bicrystal GBs, in terms of the time-correlation functions of mean-square displacement (MSD) and non-Gaussian parameter (NGP). But on the other hand, these GB atoms also demonstrate some different dynamic behaviors due to the more complicated microstructure. In particular, the immobile GB atoms are localized into their equilibrium positions due to the strong cage effect, and they are mainly at the surfaces of grains, especially the large grains, while the mobile GB atoms are active and can easily hop away from the cage around them. These mobile GB atoms gather together at the surfaces of small grains and the triple junction regions of GBs, and they form some abnormally big clusters. The size distribution of these clusters significantly deviates from the power law. Ó 2015 Elsevier B.V. All rights reserved.
1. Introduction Bulk nanocrystalline materials have been a subject of extensive research in recent decades, due to their unique physical and chemical properties [1]. This class of materials is distinguished by a significant volume of grain boundaries (GBs). The dynamic behaviors of GB atoms, such as diffusion, migration, and sliding, play a pivotal role in the deformation and grain growth of bulk nanocrystalline metals [2–5]. Many studies [3–8] have been performed to elucidate the spatiotemporal evolution of GBs in the bulk nanocrystalline metals, but our understanding of GB dynamics at atomic level is still limited owing to the disparities in the length and time scales associated with its experimental observation [9]. The width of GBs in the bulk nanocrystalline structure usually is about 2–4 atomic layers, so it is difficult to experimentally probe its microstructure and dynamic evolution at the atom level [10]. Molecular dynamics (MD) simulation is an effective tool to study the microstructure and dynamics of bulk nanocrystalline materials, as a very useful extension of experimental and analytical investigations. Some simulation studies [11–13] showed that the microstructure of GBs in bulk nanocrystalline metals is disordered, quite similar to the glass systems. Other simulations [14–16] also ⇑ Corresponding author. E-mail address: [email protected] (Z.Y. Hou). http://dx.doi.org/10.1016/j.commatsci.2015.06.039 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.
suggested the similarities between the bicrystal GBs and the glass systems in dynamics, evidenced by the dynamic heterogeneous and the string-like cooperative atomic motion. And the glassy dynamics of GBs has been experimentally confirmed in some colloidal systems [17,18]. However, previous dynamic studies of GBs focused on the bicrystal GBs which comprise only two grains and their interfacial regions, while the atomic dynamics of GBs in the bulk nanocrystalline metals was seldom reported. The microstructure and dynamics of GBs in the realistic bulk nanocrystalline metals are more complex than those in the bicrystal GBs. For example, the GBs in the bulk nanocrystalline structure contain lots of triple junction (TJ) regions which usually have particular kinetic and thermodynamic properties [19–22]. As the nanograins in the bulk nanocrystalline structure have different sizes and random crystallographic orientation, the GB dynamics were found to vary with the misorientation angle between adjacent grains [17]. Furthermore, recent studies [23–25] demonstrated that a large percentage of GB atoms are in the regions with significant crystalline order, while the disordered amorphous-like atoms mainly locate at the TJ regions. In our previous work [25,26], the bulk nanocrystalline aluminium was produced by means of a large-scale MD simulation of the rapid solidification process of liquid, and the microstructure of GBs is investigated in detail. It was found that the
178
Z.Y. Hou et al. / Computational Materials Science 108 (2015) 177–182
microstructure features of grains and GBs obtained in our simulations are consistent with those produced in some experiments and geometrically constructed by the Voronoi cell method. In this work, we further investigate the atomic dynamics of these GBs. It is found that the GB atoms display the dynamics similar to those in the glass systems and bicrystal GBs in some aspects, but also exhibit some different dynamic features from the latter. 2. Computational methods 2.1. Molecular dynamics simulation The rapid solidification process of liquid aluminium is simulated using the LAMMPS codes [27] for a system containing 1,048,576 atoms in a cubic box with periodic boundary conditions. The interaction potential adopted here is the embedded atoms method (EAM) potential proposed by Mendelev et al. [28]. As shown in Refs. [25,26], the bulk nanocrystalline aluminium is obtained below the temperature 473 K, when the liquid Al is quenched from 1273 K at a cooling rate of 1 1012 K/s. The local atomic clusters in the system are identified in terms of a recently developed method of largest standard cluster analysis (LSCA) [29], in which the neighbor of an atom are identified with a parameter-free topological criterion rather than a fixed cut-off distance rc. The method of identified grain and GB atoms are described in Refs. [25,26]. In the nanocrystalline structure, as shown in Refs. [25,26], the nanograins with an average size of 6.4 nm are separated by high-angle GBs. The GB regions neighboring the grains display FCC-like short-to-long range order. The TJ regions of GB are not disordered like liquid, but present icosahedral- and BCC-like short-range order. To investigate the dynamic properties of GB atoms, the nanocrystalline structures at different temperatures, every 100 K from 473 K to 273 K, are respectively relaxed 500 ps under an isothermal–isochoric (NVT) ensemble. More detailed information on the simulation methods can be obtained in Refs. [25,26]. 2.2. Characterization of dynamics The mean-square displacement (MSD) [30] is usually used to study the dynamic motion of atoms, which is defined as
hr2 ðtÞi ¼
N D E 1X jri ðtÞ ri ð0Þj2 ; N i¼1
ð1Þ
Fig. 1. Time dependence of MSD for the GB atoms in the bulk nanocrystalline aluminium at different temperatures.
the short time from the beginning (t < 0.1 ps), all curves show a power-law behavior which indicates the ballistic motion of atoms. At the intermediate time, the atoms are trapped in the transient cages formed by their neighbor atoms, and the trapped atoms need some time to escape from the cages. So a plateau appears on the MSD curve due to the cage effect, and it becomes pronounced with the decrease of temperature. At the final time, the curves increase with the slope n < 1, indicating the onset of cage-breaking rearrangement. The time window correlated with the cage effect is usually called the b-relaxation regime and the following time window is the a-relaxation regime [30]. To further assess the dynamics of GB atoms in the bulk nanocrystalline aluminium, its dynamic heterogeneity is investigated by means of the NGP. As shown in Fig. 2, when t < 0.1 ps, all NGPs at different temperatures are close to zero, indicating the Gaussian distribution of the vibration displacements of GB atoms. During the following b-relaxation window, all NGP curves rise monotonically to a maximum value, and the position of maximum shifts toward the longer time as the temperature decreases. This suggests that the GB atoms in the bulk nanocrystalline aluminium exhibit dynamic heterogeneity in the b-relaxation window, and such heterogeneity increases with the enhancement of cage effect.
where N is the number of atoms in the system, ri(t) is the position vectors of the ith atom at time t. The non-Gaussian parameter (NGP) [31] is frequently used to quantify the dynamic heterogeneity of supercooled liquids, and is defined as:
a2 ðtÞ ¼
3hr 4 ðtÞi 5hr 2 ðtÞi2
1;
ð2Þ
where hr4(t)i is the mean quartic displacement. The heterogeneity of the atom motion can thus be inferred from the time evolution of a2, which is zero for a Gaussian distribution, and a2 > 0 for non-Gaussian behavior. The a2 reaches a maximum for a characteristic time named t*. 3. Results and discussion 3.1. Dynamics of grain boundaries Fig. 1 shows the time dependence of MSD for the GB atoms in the bulk nanocrystalline aluminium at different temperatures. In
Fig. 2. Time dependence of non-Gaussian parameter (NGP) for GB atoms in the bulk nanocrystalline aluminium at different temperatures.
Z.Y. Hou et al. / Computational Materials Science 108 (2015) 177–182
179
Fig. 3. Probability distribution of displacements of GB atoms P(Dx) over Dt = t⁄ in the bulk nanocrystalline aluminium at 273 K. The solid line is the Gaussian fit. Particles outside the shaded region are the top 5% most mobile GB atoms.
Fig. 6. Time dependence of MSD for the mobile and immobile GB atoms, all GB atoms and grain atoms in the bulk nanocrystalline aluminium at 273 K.
Fig. 4. Distributions of mobile and immobile GB atoms around the grains with different sizes.
Fig. 7. Probability distribution P(n) of cluster size n for the mobile GB atoms in the nanocrystalline aluminium at 473 K and 273 K.
The MSD and NGP of GBs in the bulk nanocrystalline aluminium are similar to those characterized in the metal glass aluminium [32]. This demonstrates the glassy dynamics of GBs in the nanocrystalline structure. However, its b-relaxation regime is more distinct than those in the colloidal polycrystals [17,18]. This may be caused by the strong cohesive interactions existing in the metal systems which would result in slow dynamics [33]. 3.2. String-like cooperative motion
Fig. 5. A cross section of the nanocrystalline aluminium (at 273 K) through its center. The grain atoms are presented in gray dots; the GB atoms are presented in gray spheres, in which the mobile and immobile GB atoms are respectively displayed in cyan and purple colours. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Cooperative motion is one of the most characteristic features of the dynamics of supercooled liquids and glasses [32–36]. Particularly, the string-like cooperative motion of mobile atoms has been observed in the atomic dynamics of supercooled liquids and glasses [34–36], GB self-diffusion [37], and interfacial dynamics of nanoparticles [38]. These cooperative motion regions usually exhibit a power size distribution, and their average extent grows upon cooling [34–37]. To investigate the cooperative dynamics of GB atoms in the bulk nanocrystalline aluminium, the mobile atoms in the GBs are identified firstly. Fig. 3 shows the probability distribution of displacements of GB atoms P(Dx) over Dt = t⁄ at 273 K. It can be found that 95% GB atoms display diffusive dynamics, while the top 5% mobile GB atoms show non-Gaussian behavior. The proportion of GB atoms with non-Gaussian behavior is smaller than that observed in the
180
Z.Y. Hou et al. / Computational Materials Science 108 (2015) 177–182
Fig. 8. String-like cooperative motion of clusters combined of mobile GB atoms in the bulk nanocrystalline aluminium at 273 K. The atom radii in the three clusters are different for better visualization. The atoms in each clusters at t = 0 and t⁄ are respectively displayed in pink and cyan colours. The arrows in the clusters indicate the displacements of cluster atoms between t = 0 and t⁄, and they are plotted being based on the magnitudes and directions of their displacements. (a) Schematic configuration of a typical cluster. (b) Square displacements of the cluster atoms in (a) during the relaxation process. (c) Schematic configuration of another typical cluster with two ‘‘strings’’ indicated by the arrows with different colours. (d) Snapshot of the biggest cluster in the system. It is combined by many different ‘‘strings’’. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Z.Y. Hou et al. / Computational Materials Science 108 (2015) 177–182
colloidal polycrystal (10%) [17], but is the same as the supercooled liquids and glasses [32–36]. So we select the 5% GB atoms with the largest displacements as the mobile atoms, and the 5% GB atoms with the smallest displacements as the immobile atoms, just as the method proposed by Donati [34]. Two mobile atoms are considered to belong to a same cluster if they are neighbors. For the mobile GB atoms, 55.1% of them are located at the nearest-neighbor coordination shell around grains, and 35.2% are at the TJ GB regions. Furthermore, the proportion of mobile GB atoms around the small grains is notably higher than that around the large grains (see Fig. 4). For the immobile GB atoms, 84.9% of them are located at the nearest-neighbor coordination shell around grains, while only 6.4% are at the TJ GB regions. And the proportion of immobile GB atoms around the large grains is slightly higher than that around the small grains. From the cross section of the nanocrystalline structure (at 273 K) through its center, as shown in Fig. 5, it can also be found that the mobile GB atoms are mainly located at the surfaces of small grains and the TJ GB regions, while the immobile GB atoms are mainly at the surfaces of grains, especially the large grains. This results are consistent with the facts that the GB atoms at the surfaces of small grains and the TJ GB regions diffuse faster than others [19–22,39,40]. Fig. 6 further shows the differences of dynamic behaviors of different GB atoms. It can be found that the plateau in the MSD curve for the immobile atoms is close to a horizontal line and the cage-breaking rearrangement are not occurred, which is similar to that for the grain atoms. While the plateau in the MSD curve for the mobile atoms is less obvious due to the weak cage effect. This indicates that the immobile GB atoms are localized into their equilibrium positions due to the strong cage effect, while the mobile GB atoms are active and can easily hop away from the cage around them. Fig. 7 shows the size distribution of the clusters combined by the mobile GB atoms. It can be found that the average size of clusters increases with decreasing temperature, which is consistent with that in the supercooled liquids and glass systems [34–36]. It can also be found that most of these clusters have a small size (n < 40), and their probability distribution P(n) decreases with n in an power law as also observed in the supercooled liquids and glasses. But P(n) for the big clusters (n > 40) significantly deviates from the power law, the sizes of these clusters are abnormally larger than others comparing with that in the glass systems and bicrystal GBs. This means that most mobile GB atoms in the nanocrystalline aluminium gather together and form some big clusters (see Fig. 5). Fig. 8(a) shows the schematic configuration of an arc-shaped string-like cluster combined by the mobile GB atoms. All cluster atoms vibrate in small amplitudes during the initial relaxation stage (see Fig. 8(b)). Then their square displacements rapidly increase at about the same time t 20 ps. This indicates that all cluster atoms hop together away from their original positions at t 20 ps, and then diffuse along the string indicated by the arrows in Fig. 8(a), showing a cooperative motion in a string-like manner. Similarly, the cluster in Fig. 8(c) also displays a string-like cooperative motion of atoms, while it is composed of two different ‘‘strings’’ indicating by the arrows with different colours. The biggest cluster found in the system is shown in Fig. 8(d), which contains a lot of ‘‘strings’’ but also exhibits a string-like cooperative motion.
4. Conclusions To investigate the atomic dynamics of GBs in the bulk nanocrystalline structure, the bulk nanocrystalline aluminium was produced by molecular dynamics simulation of the solidification
181
process of a large-scale system, and the dynamic properties of GB atoms have been characterized and investigated by various methods. It is found that the time-correlation functions of MSD and NGP for the GB atoms in the bulk nanocrystalline aluminium display the similar features as those in the metal glass aluminium and bicrystal GBs. In particular, the MSD curve shows a pronounced plateau at the intermediate time due to the cage effect; the NGP curve rise monotonically to a maximum value in the b-relaxation window; and the mobile GB atoms undergo the cooperative motion in a string-like manner. These dynamic features become more pronounced with decreasing temperature. On the other hand, the GB atoms in the nanocrystalline structure have some different dynamic features comparing with those in the glass systems and bicrystal GBs. In particular, the atoms in the different regions of GB display notably different dynamic behaviors. The immobile GB atoms are localized into their equilibrium positions due to the strong cage effect, and they are mainly at the surfaces of grains, especially the large grains, while the mobile GB atoms are active and can easily hop away from the cage around them. The mobile GB atoms in the bulk nanocrystalline structure gather together at the surfaces of small grains and the TJ GB regions, and form some abnormally big clusters comparing with those in the glass systems and bicrystal GBs. The size distribution of the mobile GB clusters satisfies a power law for the small clusters with size n < 40, but significantly deviates from the power law for large clusters. Acknowledgments The authors would like to acknowledge the support provided by the National Natural Science Foundation of China (Grant Nos. 51101022 and 51071065). This work is also supported by the Fundamental Research Funds for the Central Universities of China (Grant Nos. 310812152001 and 2013G1121082). References [1] C.C. Koch, J. Mater. Sci. 42 (2007) 1403. [2] B.N. Kim, K. Hiraga, Y. Sakka, B.W. Ahn, Acta Mater. 47 (1999) 3433. [3] A.J. Haslam, V. Yamakov, D. Moldovan, D. Wolf, S.R. Phillpot, H. Gleiter, Acta Mater. 52 (2004) 1971. [4] Z. Shan, E.A. Stach, J.M.K. Wiezorek, J.A. Knapp, D.M. Dollstaedt, S.X. Mao, Science 305 (2004) 654. [5] S.V. Bobylev, N.F. Morozov, I.A. Ovid’Ko, Phys. Rev. Lett. 105 (2010) 0555504. [6] V. Yamakov, D. Moldovan, K. Rastogi, D. Wolf, Acta Mater. 54 (2006) 4053. [7] D. Jang, M. Atzmon, J. Appl. Phys. 99 (2006) 083504. [8] Y.T. Zhu, X.Z. Liao, X.L. Wu, Prog. Mater Sci. 57 (2012) 1. [9] S. Gokhale, K.H. Nagamanasa, R. Ganapathy, A.K. Sood, Soft Matter 9 (2013) 6634. [10] H. Van Swygenhoven, Science 296 (2002) 66. [11] S.R. Phillpot, D. Wolf, H. Gleiter, J. Appl. Phys. 78 (1995) 847. [12] P. Keblinski, S.R. Phillpot, D. Wolf, H. Gleiter, Phys. Rev. Lett. 77 (1996) 2965. [13] J. Löffler, J. Weissmüller, Phys. Rev. B 52 (1995) 7076. [14] H. Zhang, D.J. Srolovitz, J.F. Douglas, J.A. Warren, Proc. Natl. Acad. Sci. USA 106 (2009) 7735. [15] H. Zhang, D.J. Srolovitz, J.F. Douglas, J.A. Warren, Phys. Rev. B 74 (2006) 115404. [16] H. Zhang, M.I. Mendelev, D.J. Srolovitz, Acta Mater. 52 (2004) 2569. [17] K.H. Nagamanasa, S. Gokhale, R. Ganapathy, A.K. Sood, Proc. Natl. Acad. Sci. USA 108 (2011) 11323. [18] T.O.E. Skinner, D.G.A.L. Aarts, R.P.A. Dullens, J. Chem. Phys. 135 (2011) 124711. [19] M.R. Chellali, Z. Balogh, H. Bouchikhaoui, R. Schlesiger, P. Stender, L. Zheng, G. Schmitz, Nano Lett. 12 (2012) 3448. [20] T. Frolov, Y. Mishin, Phys. Rev. B 79 (2009) 174110. [21] M.R. Chellali, Z. Balogh, L. Zheng, G. Schmitz, Scripta Mater. 65 (2011) 343. [22] O.K. Johnson, C.A. Schuh, Acta Mater. 61 (2013) 2863. [23] H. Van Swygenhoven, D. Farkas, A. Caro, Phys. Rev. B 62 (2000) 831. [24] Z. Wu, Y. Zhou, X. Zhang, S. Wei, D. Chen, Appl. Phys. Lett. 84 (2004) 4442. [25] Z. Hou, Z. Tian, Y. Mo, R. Liu, Comput. Mater. Sci. 92 (2014) 199. [26] Z. Hou, Z. Tian, R. Liu, K. Dong, A. Yu, Comput. Mater. Sci. 99 (2015) 99. [27] S.J. Plimpton, J. Comp. Phys. 117 (1995) 1. [28] M.I. Mendelev, M.J. Kramer, C.A. Becker, M. Asta, Phil. Mag. 88 (2008) 1723. [29] Z.A. Tian, R.S. Liu, K.J. Dong, A.B. Yu, EPL 96 (2011) 36001. [30] W. Kob, J. Phys.: Condens. Matter 11 (1999) R85.
182 [31] [32] [33] [34]
Z.Y. Hou et al. / Computational Materials Science 108 (2015) 177–182
A. Rahman, Phys. Rev. 136 (1964) A405. M. Li, C.Z. Wang, M.I. Medelev, K.M. Ho, Phys. Rev. B 77 (2008) 184202. Z. Zhang, P.J. Yunker, P. Habdas, A.G. Yodh, Phys. Rev. Lett. 107 (2011) 208303. C. Donati, S.C. Glotzer, P.H. Poole, W. Kob, S.J. Plimpton, Phys. Rev. E 60 (1999) 3107. [35] E.R. Weeks, J.C. Crocker, A.C. Levitt, A. Schofield, D.A. Weitz, Science 287 (2000) 627.
[36] [37] [38] [39] [40]
Y. Gebremichael, M. Vogel, S.C. Glotzer, J. Chem. Phys. 120 (2004) 4415. Y. Mishin, Defect Diffus. Forum 363 (2015) 1. H. Zhang, P. Kalvapalle, J.F. Douglas, Soft Matter 6 (2010) 5944. B. Arab, S. Sanjabi, A. shokuhfar, Mater. Lett. 65 (2011) 712. D. Prokoshkina, V.A. Esin, G. Wilde, S.V. Divinski, Acta Mater. 61 (2013) 5188.