PHYSICAL REVIEW B 84, 092201 (2011)
Origin of splitting of the second peak in the pair-distribution function for metallic glasses S. P. Pan, J. Y. Qin,* W. M. Wang, and T. K. Gu Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials (Ministry of Education), Shandong University, Jinan 250061, China (Received 24 May 2011; revised manuscript received 8 August 2011; published 12 September 2011) The connection of atomic clusters has been investigated using molecular-dynamics simulation to explain the splitting of the second peak of the pair-distribution function in amorphous alloys. It is found that the unevenness of the connecting style of atomic clusters results in the splitting of the second peak and the two subpeaks are caused by a three-atom-shared connection and a one-atom-shared connection between atomic clusters. The underlying reason is that metallic glasses have higher density than liquid alloys and a different connecting style of atomic clusters from crystals. DOI: 10.1103/PhysRevB.84.092201
PACS number(s): 61.43.Dq
Understanding the atomic structure of metallic glasses (MGs) is a challenging but interesting topic in material science and physics.1–6 Several early models, such as the dense random packing of hard spheres7 and the trigonal prism model,8 and other studies9,10 made significant progress toward our understanding of short-range order (SRO) in the atomic structure of MGs. However, they are still insufficient for our understanding.11 Recent models focus on medium-range order (MRO) by proposing different cluster-packing schemes, such as efficient cluster packing on a face-centered-cubic lattice12 and quasiequivalent clusters on an icosahedral packing.13 In these two models, clusters are connected by sharing common atoms in their shells. The connection of clusters brings a bridge for short- to medium-range order in MGs and is very helpful for our understanding of MRO. However, few works have been done from a statistical point of view. It is known that the atomic structure of MGs is similar to that of liquid alloys but have splitting in the second peak of the pair-distribution function (PDF), which is regarded as a characteristic indication of MGs. The splitting reflects the atomic packing beyond the nearest neighbors and also the characteristics of MRO. However, neither of the two models can give an unambiguous reason for the splitting. The origin of splitting in the second maximum of the PDF for MGs has been investigated for many years.14–18 Huerta et al. explained the second peak splitting of the PDF in a two-dimensional (2D) system of hard disks by a loose crystalline hexagonal ordering with a proper gap width between the Voronoi nearest neighbors (VNNs) and the farther and closer next-VNNs’ contribution to the two subpeaks.15 With respect to the clusters, the farther and closer next-VNNs can be regarded as the central disks of the clusters sharing vertices and edges with the cluster centered by the central disk. More connecting styles between clusters would be involved in 3D systems. Bennett introduced several typical connected groups of particles in hard contact to explain the splitting.16 It seems that the positions of subpeaks in this model agree well with experimental data. However, another peak suggested in this model was not detected in simulations and experiments. In fact, these simple groups indicate different connecting styles of atomic clusters, which implies that there might be some correlation between the splitting and the connection of atomic clusters. However, atomic clusters are not simply connected in 1098-0121/2011/84(9)/092201(4)
hard contact in realistic systems, which may be the reason for the absence of the third suggested peak. Therefore, it is necessary to investigate the connection of atomic clusters statistically in realistic 3D systems. Voloshin and Naberukhin pointed out that the splitting is formed of chains in which all Delaunay simplices are nearly regular tetrahedra and quartoctahedra.17 However, not all the atoms can be involved in good simplices in liquid and amorphous alloys. A Voronoi polyhedron is usually used to characterize SRO; therefore, explaining the splitting by Delaunay simplices may break the correlation between SRO and MRO. Moreover, neither of these models gives any further reason for the splitting. Metallic glasses are usually obtained by supercooling the melts and restraining crystallization. Therefore, to understand the second peak splitting of the PDF in MGs it is necessary to investigate the structural evolutions of liquid-glass and liquidcrystal transitions. In this Brief Report pure Fe is chosen as a model to study the two transitions by molecular-dynamics (MD) simulation. The liquid-glass transition for the Cu50 Zr50 alloy is also investigated for further verification. Our aim is to investigate the close correlation between the splitting of the second peak of the PDF in MGs and the connection of clusters. A large-scale atomic/molecular massively parallel simulator19 with an embedded-atom method potential20,21 is used to carry out the MD simulation for pure Fe and Cu50 Zr50 alloy. NPT ensembles (P = 0) of pure Fe and Cu50 Zr50 alloy containing 30 000 atoms with periodic boundary conditions are melted and equilibrated at 2000 K for 1 ns (time step: 1 fs) and cooled down to 300 K at different cooling rates. To test the effect of the initial quench configurations and quench rates on the glass transition, we quench six different configurations of Fe from 2000 to 300 K and quench Cu50 Zr50 alloy with three different cooling rates. A Voronoi polyhedron analysis22 is performed to characterize SRO in liquid alloys and MGs. The Voronoi polyhedral index is expressed as (n3 , n4 , n5 , n6 , . . .), where ni denotes the number of i-edged faces of the Voronoi polyhedron. The small faces are removed if their areas are smaller than 10% of the average area of all the faces of the polyhedron. The PDF is also used to describe the structure and then decomposed according to the status of the clusters and their connections. Figure 1 displays PDFs of liquid, amorphous, crystalline, and ideal crystalline Fe, respectively. First, we illustrate the
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FIG. 2. (Color online) Decomposition of the PDF with standard deviation for amorphous Fe.
FIG. 1. (Color online) Pair-distribution functions of liquid, amorphous, crystalline, and ideal crystalline Fe: black line, liquid Fe (2000 K); red line, amorphous Fe (cooling rate: 1013 K/s); blue line, crystalline Fe (cooling rate: 1011 K/s); and purple line, ideal crystalline Fe.
correspondence of the peaks of the PDF for ideal crystalline Fe with respect to the atomic clusters. r0 is the position of the centered atom and ri refers to the position of the ith peak. It is known that the Voronoi index of SRO around an Fe atom in ideal body-centered-cubic crystal is 0,6,0,8. The atoms at r1 and r2 are the shell atoms of 0,6,0,8. The polyhedron around the atom at r3 shares four atoms in the shell with the polyhedron around the atom at r0 . The number of shared atoms for the polyhedron around the atom at r0 and polyhedra around atoms at r4 , r5 , and r6 are 2, 1, and 1, respectively. Beyond r6 , no connection exists. Second, we find that the positions of the peaks of the PDF for crystalline Fe agree well with those for ideal crystalline Fe, which indicates that the former have correspondences that are similar to those of the latter. Finally, the first peak of g(r) for amorphous Fe is at almost the same position as that for ideal crystalline Fe. However, polyhedra in amorphous Fe are quite different from those in ideal crystalline Fe. There are more than one hundred kinds of polyhedra and five have a fraction over 5% in amorphous Fe. They are 0,3,6,4,0,1,10,2,0,2,8,4,0,0,12,0, and 0,1,10,3. All these polyhedra have a dominant number of pentagons while the 0,6,0,8 polyhedron in ideal crystalline Fe has no pentagons. The second maximum of the PDF for amorphous Fe is split into two subpeaks whose positions are near peaks correlated with the polyhedral connection for ideal crystalline Fe. Therefore, the question arises as to whether the two subpeaks of the second maximum in g(r) for amorphous Fe also have some correlation with the polyhedral connection. To address the problem, we decompose g(r) in such a way: Fe(s) is used to stand for the shell atoms of the polyhedron around the atom at r0 and Fe(c) refers to the atoms not Fe(s) and whose polyhedra share common atoms with the
polyhedra around the objective atom; Fe(c-i) belongs to Fe(c) and i denotes the number of common atoms of the polyhedral connection. Figure 2 demonstrates the decomposition of g(r) with standard deviation for amorphous Fe. Since the standard deviation is too small, the decomposition of g(r) is insensitive to the initial quench configurations. We find that gFe-Fe(s) (r) and gFe-Fe(c) (r) construct the first and second maxima of gFe-Fe (r) perfectly, which indicates that the splitting of the second peak is closely correlated with the polyhedral connection. Then gFe-Fe(c) (r) is further decomposed according to the number of common atoms of the polyhedral connection. It is amazing that all four gFe-Fe(c-i) (r) curves are nearly symmetrical and almost can be described by the Gaussian functions shown in Fig. 3. Among the curves of gFe-Fe(c-i) (r), gFe-Fe(c-3) (r) and gFe-Fe(c-1) (r) have the two highest peaks. The peak of gFe-Fe(c-2) (r) is so low that when all the gFe-Fe(c-i) (r) curves are composed into the second peak of g(r), the second peak seems to be separated into two subpeaks. The unevenness of the style of polyhedral connection leads to the splitting of the second peak of g(r) for amorphous Fe. We classify the polyhedral connection beyond the nearest neighbors into four groups according to the number of common atoms. The one-atom-shared connection is a vertex-shared connection, the two-atom-shared connection corresponds to an edge-shared connection, and the three-atom-shared and four-atom-shared connections are face-shared connections. Bennett introduced three typical connected groups of particles to explain the splitting.16 These three connected groups indicate three connecting styles of atomic clusters, which are one-atom-shared, two-atom-shared, and three-atomshared connections. Bennett proposed that the first subpeak at 1.732R1 (R1 is the position of the first peak of the PDF) results from the two-atom-shared connection, the second subpeak at 2R1 originates from the one-atom-shared connection, and there should be another subpeak at 1.633R1 caused by the three-atom-shared connection. The typical particles in this model are connected in hard contact; however, the realistic polyhedral connection is not hard contact. For amorphous Fe, the positions of the peaks of the four-atom-shared, three-atomshared, two-atom-shared, and one-atom-shared connections
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FIG. 3. (Color online) Gaussian fitting of gFe-Fe(c-i) (r) (i = 1–4): (a) gFe-Fe(c-1) (r), (b) gFe-Fe(c-2) (r), (c) gFe-Fe(c-3) (r) and (d) gFe-Fe(c-4) (r).
˚ respectively. Their proportions are 3.87, 4.23, 4.70, and 5.19 A, ˚ are 1.57, to the position of the first peak in g(r) (2.47 A) 1.71, 1.90, and 2.1, which are different from those proposed by Bennett. Our results suggest that the first subpeak results from the three-atom-shared connection, the second one mainly originates from the one-atom-shared connection, and the hollow between the two subpeaks is caused by the smaller population of two-atom-shared connections. We investigate the atomic structure of Cu50 Zr50 MG for further verification. Figure 4(a) shows a comparison of PDFs for Cu50 Zr50 MG obtained with three different cooling rates. It is clear that the PDFs are not sensitive to the cooling rate. Therefore, although the quench rates used in our simulations are orders of magnitudes faster than those in the experiments
FIG. 4. (Color online) (a) Comparison of PDFs for Cu50 Zr50 MG obtained with three cooling rates and decomposition of PDFs (b) gCu-Zr (r), (c) gZr-Zr (r), and (d) gCu-Cu (r) for Cu50 Zr50 MG obtained with a cooling rate of 1013 K/s.
FIG. 5. (Color online) (a) Decomposition of g(r) for liquid Fe (2000 K), (b) intensity of gFe-Fe(c-i) (r) (i = 1–3) as temperature decreases, (c) average number of i-atom-shared connections (i = 1–3) around an atom as temperature decreases, and (d) decomposition of g(r) for crystal Fe (obtained with a cooling rate of 1011 K/s).
in the laboratory, our results obtained by simulations should be appropriate and reliable. Figures 4(b)–4(d) display the decomposition of partial PDFs for Cu50 Zr50 MG obtained with a cooling rate of 1013 K/s, which also suggests that the unevenness of the connecting style of atomic clusters results in the splitting of the second peak. Although amorphous Fe and Cu50 Zr50 MG are two quite different systems with different atomic interactions and atomic radii of the constituting atoms, they show the same origin for the second peak splitting of the PDF. Moreover, besides the total PDF, Figs. 4(b)–4(d) suggest that our theory is also applicable to the partial PDF. These facts indicate that our conclusion has wide feasibility. To determine the reason for the unevenness of the connecting style we compare the decompositions of g(r) for liquid and amorphous Fe. It can be seen from Fig. 5(a) that the intensities of the peaks of gFe-Fe(c-1) (r), gFe-Fe(c-2) (r), and gFe-Fe(c-3) (r) for liquid Fe are nearly the same. Figure 5(b) shows that the intensity of gFe-Fe(c-3) (r) grows more markedly than the intensities of gFe-Fe(c-2) (r) and gFe-Fe(c-1) (r). The reason for this situation is explained as follows. It is known that density increases as temperature decreases for most of the systems. The number of face-shared connections should increase if the density of the system increases. Voloshin and Naberukhin pointed out that all Delaunay simplices in liquid alloys and metallic glasses are nearly regular tetrahedra and quartoctahedra,17 which means that most of the faces of a polyhedron (named a coordination polyhedron) whose vertices are at the shell of a Voronoi polyhedron are triangles and some of them might be quadrilateral. Therefore, as the temperature decreases, three-atom-shared connections in liquid alloys are enriched in order to enhance the density, which is shown in Fig. 5(c). Figure 5(c) also shows that the average number of two-atom-shared connections around an atom decreases as the temperature decreases. We will give the reason in the following. Figure 1 shows that the numbers of common atoms of a polyhedral connection in ideal crystal Fe are 4, 2, and 1, which indicates that there is no three-atom-shared
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connection. The decomposition of g(r) for crystalline Fe by supercooling Fe melts with a cooling rate of 1011 K/s in Fig. 5(d) also indicates that four-atom-shared and two-atomshared connections are two dominant styles of polyhedral connection in crystalline Fe. During the liquid-glass transition, four-atom-shared and two-atom-shared connections should be restrained to avoid crystallization, which we think is the reason for the decrease in the number of two-atom-shared connections. The number of one-atom-shared connection stays almost unchanged as the temperature decreases, as shown in Fig. 5(c). Therefore, an additional reason for the splitting of the second peak of the PDF for MG is that MG has a larger density than the liquid state and different connecting styles of clusters from the crystal state. We now look at the second peak splitting in a 2D system of hard disks in Fig. 1 of Ref. 15. There are only two kinds of connecting styles between clusters in a 2D system, edge- and vertex-shared connections, which results in the two subpeaks of the second peak in the PDF. A higher density necessitates
more edge-shared connections in 2D systems. That is why the left shoulder gradually transforms into a subpeak as the density increases. These facts indicate that there is also some close correlation between the second peak splitting and the connection of clusters in 2D systems. In summary, we investigated the connection of short-range order in liquids and glasses and explained the splitting of the second peak of the PDF for the glass state. The results demonstrate that the splitting is caused by the unevenness of polyhedral connecting styles in the glass state, which results from its higher density than liquid and different style of cluster dense packing from crystal.
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[email protected] A. L. Greer and E. Ma, MRS Bull. 32, 611 (2007). 2 Y. Q. Cheng, E. Ma, and H. W. Sheng, Phys. Rev. Lett. 102, 245501 (2009). 3 T. Fujita, K. Konno, W. Zhang, V. Kumar, M. Matsuura, A. Inoue, T. Sakurai, and M.W. Chen, Phys. Rev. Lett. 103, 075502 (2009). 4 A. Hirata, P. F. Guan, T. Fujita, Y. Hirotsu, A. Inoue, A. R. Yavari, T. Sakurai, and M. W. Chen, Nature Mater. 10, 28 (2011). 5 M. Z. Li, C. Z. Wang, S. G. Hao, M. J. Kramer, and K. M. Ho, Phys. Rev. B 80, 184201 (2009). 6 H. L. Peng, M. Z. Li, and W. H. Wang, Phys. Rev. Lett. 106, 135503 (2011). 7 J. D. Bernal, Nature (London) 183, 141 (1959). 8 P. H. Gaskell, Nature (London) 276, 484 (1978). 9 V. A. Borodin, Philos. Mag. A 79, 1887 (1999). 10 Y. Q. Cheng and E. Ma, Prog. Mater. Sci. 56, 379 (2011). 1
We thank Maozhi Li from Renmin University and Limin Wang from Yanshan University for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 50971082 and 50831003 to J.Y.Q., Grant No. 10974117 to T.K.G., and Grant No. 51171091 to W.M.W.).
D. B. Miracle, T. Egami, K. M. Flores, and K. F. Kelton, MRS Bull. 32, 629 (2007). 12 D. B. Miracle, Nature Mater. 3, 697 (2004). 13 H. W. Sheng, W. K. Luo, F. M. Alamgir, J. M. Bai, and E. Ma, Nature (London) 439, 419 (2006). 14 J. L. Finney, Proc. R. Soc. London Ser. A 319, 479 (1970). 15 A. Huerta, D. Henderson, and A. Trokhymchuk, Phys. Rev. E 74, 061106 (2006). 16 C. H. Bennett, J. Appl. Phys. 43, 2727 (1972). 17 V. P. Voloshin and Y. I. Naberukhin, J. Struct. Chem. 38, 62 (1997). 18 X. J. Liu, Y. Xu, X. Hui, Z. P. Lu, F. Li, G. L. Chen, J. Lu, and C. T. Liu, Phys. Rev. Lett. 105, 155501 (2010). 19 S. Plimpton, J Comput. Phys. 117, 1 (1995). 20 M. I. Mendelev, Philos. Mag. 83, 3977 (2003). 21 M. I. Mendelev, J. Appl. Phys. 102, 043501 (2007). 22 N. N. Medvedev, J. Comput. Phys. 67, 223 (1986).
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