Dec 15, 2006 - placed on the position in proportion to the atomic size. In this study .... where r is the distance, RA, RB, and RC are the atomic radii of the A atom ...
Materials Transactions, Vol. 47, No. 12 (2006) pp. 2904 to 2909 #2006 The Japan Institute of Metals
Effects of Atomic Size for Voronoi Tessellation Technique on Binary and Ternary Systems of Metallic Glasses Junyoung Park and Yoji Shibutani Department of Mechanical Engineering, Osaka University, Osaka 565-0871, Japan A number of researchers are still employing the ordinary Voronoi tessellation technique to explore the atomic structures of metallic glasses. However, the ordinary Voronoi tessellation technique does not take into consideration the size of each atom, and this essentially leads to differences in the atomic volume and structure, such as the number of atoms with FCC or icosahedral structure. An alternative way, the radical plane technique, is applied to binary and ternary systems: such as Zr, ZrCu, ZrCuAl, ZrB, and ZrFeB. In the radical plane technique, the face is placed on the position in proportion to the atomic size. In this study, we found that the differences cannot be negligible. In some cases, the difference in the results between the two techniques is more than 50%. [doi:10.2320/matertrans.47.2904] (Received July 24, 2006; Accepted October 17, 2006; Published December 15, 2006) Keywords: metallic glasses, atomic structure, radical plane technique
1.
Introduction
Since the first synthesis of a system with amorphous structure, metallic glasses have been widely studied for a few decades now. In particular, both experimental and computational studies have focused on the structural characteristic of metallic glasses.1) Molecular dynamics (MD) and Monte Carlo modeling have been adopted to probe the internal structure of metallic glass materials in computational studies.2,3) The internal structures of amorphous materials are mainly investigated using the Voronoi tessellation technique in these computational studies, while neutron diffraction is used in experimental studies. An earlier MD study has suggested that icosahedra are a possible structural unit of amorphous materials.4) In addition, it is well known that for amorphous materials, macroscopic properties depend on the internal structure. Thus, it is important to precisely analyze the internal structure. A common technique, the ordinary Voronoi tessellation technique, was originally developed to calculate the volume of each sphere and packing of spheres by Voronoi.5) In this technique, a face of the Voronoi polyhedron is placed halfway between atoms. Then, the edges of the polyhedron made by Voronoi tessellation are formed by the intersections of these faces. Using the faces and the edges, a polyhedron for each atom can be defined. Thus, the polyhedron with a given number of faces indicates the local structure around the corresponding atom. The volume of each atom is defined as the volume of the polyhedron. Based on these techniques, a number of researchers, even in very recent papers, have adopted the ordinary Voronoi tessellation technique as an analysis tool to determine the internal structure.2,3) Similar to the internal structures of amorphous materials, in biology, the density of proteins at the atomic level needs to be carefully measured to predict the thermo-stability of cavity-filling mutants.6) However, bisection adopted by the ordinary Voronoi tessellation technique cannot depict the exact plane to represent the polyhedron because it does not consider the size of the atom. It miscalculates the volume of the protein. To overcome this error, some alternate ways have been
suggested in biology; Richards has suggested one method by considering the ratio of two atoms.7) This technique, however, still miscalculates the atomic volume, called the ‘‘vertex error’’.8) The radical plane technique suggested by Fischer and Koch (introduced by Gellatly and Finney) solves the vertex error using the plane positioned on the locus of points from which the tangent lengths to the two spheres are equal.8) Gerstein and others first introduced nonplanar boundaries, called Gerstein spheres, between atoms, such that the ratio of distances to the centers of the neighboring atoms is constant.9) However, Gerstein’s method using non-planar boundaries is difficult to be applied to the internal structures of amorphous materials because it forms a swollen polyhedron rather than a polyhedron with planar boundaries. In order to show the difference between the ordinary Voronoi tessellation technique and the radical plane technique, in the present study, two ternary amorphous systems, ZrFeB and ZrCuAl, two binary amorphous systems, ZrB and ZrCu, and, a pure Zr system with amorphous structure were adopted. These systems were used because they are relatively stable in the ab initio calculations.10) The case for the binary system based on Zirconium atoms and smaller fictitious atoms using Richard’s method has already been studied by Park and Shibutani.11) 2.
Computational Model
All models consist of 2000 atoms subject to a periodic boundary condition. The models used in this study are Zr100 , Zr65 Cu35 , Zr65 Cu25 Al10 , Zr80 B20 , and Zr40 Fe40 B20 . A molecular dynamics study using a modified Lennard-Jones potential is adopted. The Zr atom is modeled as a basic atom, termed the A atom. The other atoms, such as Cu, Al, B and Fe, termed the B and/or C atoms, are described by the modified Lennard-Jones potentials, defined by potentials similar to the Zr-Zr interactions but whose shape and position are scaled by the size of the B atoms and C atoms. For example,
Effects of Atomic Size for Voronoi Tessellation Technique on Binary and Ternary Systems of Metallic Glasses
RA B{B ðrÞ ¼ r RB RA r C{C ðrÞ ¼ RC 2 r A{B ðrÞ ¼ ð1 þ RB =RA Þ 2 r A{C ðrÞ ¼ ð1 þ RC =RA Þ 2 r B{C ðrÞ ¼ ðRB =RA þ RC =RA Þ
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ð1Þ ð2Þ ð3Þ ð4Þ ð5Þ
where r is the distance, RA , RB , and RC are the atomic radii of the A atom, the B atom and the C atom, respectively. The locations of each atom in all models are randomly chosen at the first step in the cubic cell with a size of about 3.5 nm. The system is then fully heated to 5,000 K for 235 ps, relaxed for 1,000 ps and then cooled to 300 K for 4,700 or 47 ps (pure Zr case) with periodic boundary condition and a cooling rate 1012 or 1014 K/s (pure Zr case). The ParinelloRahman method is applied to keep the pressure constant.12) Using the above conditions, all models show amorphous structure in the final stage. The radii of monotonic atom could be different with that of the composition of materials based on its local environment. However, the radii of atom used in this study are fixed. For more accurate result, the atomic radii need to be defined by some rigorous way. 3.
(a) Ordinary Voronoi
(b) Radical plane technigue
Fig. 1 Two-dimensional schematics of (a) the ordinary Voronoi tessellation technique and (b) the radical plane technique. In the ordinary Voronoi tessellation, the size of the smaller atom is overestimated. The atomic structure, i.e., the number of faces of each atom also depends on the atomic size. [From Ref. 11)]
Radical Plane Technique8,13)
In the ordinary Voronoi tessellation technique derived from a collection of points rather than spheres with some size, the entire space is partitioned among a collection of equal-sized atoms. Each face of the polyhedron is the bisecting plane of inter-atomic vectors and is perpendicular to the vectors between two atoms. Each edge of polyhedron is derived from the intersection of two planes consisting of three atoms. A vertex of the polyhedron is also formed by three edges consisting of four atoms. Thus, the vertex can be determined as a point with equal distance from four atoms. The equation to determine the location of the vertex is as follows: ðxi xv Þ2 þ ðyi yv Þ2 þ ðzi zv Þ2 ¼ L2c ;
i¼14
ð6Þ
where ðxi ; yi ; zi Þ are the coordinates of each atom, ðxv ; yv ; zv Þ are the coordinates of the vertex and Lc is the distance from the vertex to the center of any atom. There must be no atom closer than Lc in the tetrahedron consisting of four atoms. Hence, in the ordinary Voronoi tessellation technique, the locations of all neighboring atoms are the only necessary information to define the polyhedron. However, as atoms are not the same size, such as Zr (Radius ¼ 0:160 nm) and B (Radius ¼ 0:088 nm) atoms, bisection cannot depict a real face such as in Fig. 1. The problem of bisection can be solved by algorithms suggested by researchers in biological fields. Richards suggests a weighted Voronoi tessellation technique, called Richard’s B method, considering the location of the plane proportional to the atomic radii.7) However, this method fails to provide a flawless mathematical method because of the vertex error, which is due to the
Fig. 2 Two-dimensional schematics of the vertex error in Richard’s B technique. The area marked by black is the vertex error unoccupied by any polygon.
space not allocated to any polyhedron, as shown in Fig. 2. The vertex error is a result of the mismatch between the plane allocated by Richard’s B method and the real contact plane at the covalent bond. To avoid the vertex error, Gellatly and Finney have introduced an alternative method, termed the radical plane technique.8) The radical plane of two spheres is the locus of points from which the tangential lengths to the two spheres are equal, as shown in Fig. 3. The equation for the vertex has a very similar form with eq. (6) as follows: ðxi xv Þ2 þ ðyi yv Þ2 þ ðzi zv Þ2 ri2 ¼ L2t ; i¼14
ð7Þ
where, ri and Lt are the radius of each atom and the distance between the vertex and the tangential point, respectively. Similar to the ordinary Voronoi tessellation technique, in this technique, there must be no atoms with a smaller distance than Lt . Details of the algorithm can be found in Refs. 8 and 13. 4.
Results and Discussion
The models with an amorphous structure were analyzed by the ordinary Voronoi tessellation technique and the radical plane technique. Two properties, atomic volume and number of face, were measured.
J. Park and Y. Shibutani
in the volume of Zr between the two techniques is increased with the number of elements, while the volume difference of B is decreased. Although the volume does not appear to show any general trend for the number of elements, the error in volume calculation is proportional to the number of elements because of the error caused by the third element, Al and B in the ternary system. For example, although the errors caused by Zr and Cu in the binary system are almost equal to those in the ternary system, there is another error caused by Al in the ternary system. The difference reached 60% in the case of the Zr-B binary system. Thus, results based on the ordinary Voronoi tessellation technique are misleading, with a relatively large error for the atomic volume.
plane Lt r1
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Schematic of the radical plane technique.
4.1 Atomic volume In this paper, the atomic volume is defined as the assigned volume to each atom by the ordinary Voronoi tessellation technique or the radical plane technique. The atomic volume has an important role in calculating local atomic strain and local hydrostatic pressure.14) Figure 4 shows the average volume for each kind of atom measured by both techniques. In the plots, the average volume for Zr100 must be equal for any technique, and both techniques return identical results for the models with the pure element. In the case of the binary system, the volume starts to show differences. As seen in Fig. 4, the volume of Zr measured by the ordinary Voronoi tessellation technique (denoted as ‘‘Ord’’ in figures) is underestimated because of the unfair partitioning for larger atoms; the Zr atom is larger than the Cu atom. In contrast, the volume of Cu is overestimated. The Zr-B model shows larger differences between the two techniques than the Zr-Cu model because the size ratio of B to Zr of 0.55 is smaller than that of Cu to Zr, which is 0.8. For the ternary system, volume differences between the two techniques were also observed. The differences, however, depend on the materials. In the case of Zr-Cu-Al, the differences are almost the same when compared to the binary system, Zr-Cu. The reason for this is not clear at the present. It may be related to the structure or the relatively small size differences in the Zr-Cu-Al system (the radius of Zr is 0.160 nm, Cu is 0.128 nm and Al is 0.143 nm). In the case of the Zr-B-Fe system, the difference
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4.2 Number of faces 4.2.1 Pure element system To compare with the binary and ternary systems, the model with the pure Zr atom was used. The analysis results estimated by the radical plane technique are essentially identical with those of the ordinary Voronoi tessellation technique, since eq. (6) is a special case of eq. (7) with ri ¼ 0. The zero radius means partitioning for points rather than spheres, corresponding to the ordinary Voronoi tessellation technique. In Fig. 5, the numbers of atoms with the given number of faces are plotted and the peak is 14 faces. 4.2.2 Binary system Figure 6 shows the distribution of polyhedra in terms of the number of atoms with a given number of faces. As expected, the distribution of Zr-Cu shows small differences between the two analysis techniques, since the radius ratio defined by the ratio of the radius of the smaller atom to that of the larger atom is Rratio ¼ rCu =rZr ¼ 0:8. The peaks of the Zr and Cu distributions also maintain the same number, 15 faces and 11 faces for Zr and Cu, respectively, for both techniques. The area of the overlapping region between Zr and Cu in the distribution measured by the ordinary Voronoi tessellation technique is reduced at the distribution measured by the radical plane technique. For the Zr-B cases, the distribution shows distinguishable differences between the two analysis techniques. While the distribution for Zr maintains the same position and shape, the distribution for B moves to the left
Averaged Volume (nm3 )
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Materials
(a) The averaged volume of each kind
(b) The averaged volume of each kind
of atom on the Zr-Cu-Al models
of atom on the Zr-Fe-B models
The averaged volume of each kind of atom as measured by both techniques for (a) Zr-Cu-Al models and (b) Zr-Fe-B models.
Effects of Atomic Size for Voronoi Tessellation Technique on Binary and Ternary Systems of Metallic Glasses 1000
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which indicates a small number of faces and changes its shape to be sharp. Similar to the Zr-Cu cases, the overlapping area between Zr and B in Fig. 6(c) almost vanishes in Fig. 6(d). This means that the polyhedron made by each atom clearly depends on the kind of atom. 4.2.3 Ternary system To determine how the number of elements affects the polyhedron distribution, the ordinary Voronoi tessellation 1000
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technique and the radical plane technique were also applied to two ternary systems, Zr65 Cu25 Al10 and Zr40 Fe40 B20 , as shown in Fig. 7. The distributions measured by the ordinary Voronoi tessellation technique in the Zr-Cu-Al system, is analogous to the distributions obtained by the radical plane technique. Like the binary systems, this is due to relatively similar atom sizes. However, it shows some differences when the details are considered. Further details will be discussed in the next section. For example, the peak of the distribution for Al moves to the left and the area of the overlapping region between Zr and Cu is reduced, compared to the case of the ordinary Voronoi tessellation technique. For the Zr-Fe-B system, the distribution of B moves to the left. This means that the polyhedron made by the B atom has a smaller face than expected. In the distributions measured by the ordinary Voronoi tessellation technique, the atoms with 12 faces correspond to an icosahedron consisting of 80% Fe and 20% B atoms. However, in the distribution of the radical plane technique, almost all of the atoms are Fe. This shows that the results strongly depend on the analysis technique when the size of the atoms in the material has differences. The peak of Fe and Zr is also changed in the radical plane technique. Consequently, although the overall shape of the distributions keeps analogy, the composition of the atoms with the given number of faces can be different. 1000
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Fig. 6 The distributions of polyhedra for binary systems, Zr65 Cu35 and Zr80 B20 , are described by the number of atoms with the given number of faces.
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Fig. 7 The distributions of polyhedra for ternary systems, Zr65 Cu25 Al10 and Zr40 Fe40 B20 , are described by the number of atoms with the given number of faces.
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Fig. 8 The differences in polyhedra distribution for (a) Zr65 Cu25 Al10 and (b) Zr40 Fe40 B20 are represented in terms of percentage, as defined by (Number by Radical plane method. —Number by Ordinary Voronoi technique)/Radical plane method.
4.2.4 Differences Figure 8 depicts the differences in the polyhedron distributions for all systems from 10-faces to 19-faces in terms of percentage. Here, the percentage is defined as the number of atoms obtained by the radical plane technique minus that by
the ordinary Voronoi tessellation technique divided by the number of atoms by the radical plane technique. Note that the number of atoms is the total number of atoms with the given number of faces, not the specified atoms, such as Zr, Cu, Fe, Al, or B. The distributions of each kind of atom are different
Effects of Atomic Size for Voronoi Tessellation Technique on Binary and Ternary Systems of Metallic Glasses
from the total number of atoms. The pure Zr cases have zero percentage difference between the two techniques as shown at the previous figures. For 10-faces and 18-faces of Zr-Cu-Al (Fig. 8(a)), the percentage is over 100%. This is not a really large difference because the denominator, the number of atoms by the radical plane technique, in terms of percentage is also small. Except for these faces, the biggest differences are for the 13-faces and 14-faces cases, typical of icosahedron-like geometry, regardless of the binary or ternary system. Although the 12-faces case, also related to the icosahedron, shows a small percentage, the difference corresponding to about 25% for 13-faces and 14-faces suggests that the results analyzed by the ordinary Voronoi tessellation technique misleads the distribution of the polyhedron. For the case of Zr-Fe-B, the biggest difference, corresponding to around 75%, is the 12-faces case of the binary system except for the cases with a small denominator. This amount of error could affect the analysis results. Although Figures 8(a) and (b) do not show any constant trend, the Zr-Fe-B system with bigger size differences shows a larger error. Unlike the volume distribution, the distribution of faces does not show a general trend. However, the error is not negligible so that the number of faces also should be analyzed by the method considering the size of the atom. 5.
Conclusions
The atomic volume and the internal structure of the binary and ternary amorphous system consisting of Zr, Cu, Fe, Al, and B, are studied by two techniques: the ordinary Voronoi tessellation technique and the radical plane technique. The averaged volume by the radical plane technique shows up to a 60% difference compared to the ordinary Voronoi tessellation technique. The error increases as the number of elements increases. For the internal structure, the frequency of 12-, 13and 14-faces, typical icosahedron-like geometry, shows up to a 92% difference. Clearly, these differences are not negli-
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gible. In addition, the composition of elements with the given number of faces can be changed. For example, for the Zr-FeB system, the composition, 80% Fe and 20% B as measured by the ordinary Voronoi tessellation technique changes to 99% Fe and 1% B measured by the radical plane technique. Although the differences between the two techniques do not show a general tendency for the number of elements, the size of the atom in the analysis of the internal structure is an essential consideration. Acknowledgements We gratefully acknowledge support from the Ministry of Education, Culture, Sports, Science and Technology in Japan, Grant-in-Aid for Scientific Research on Priority Areas (15074214).
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