液態(tài)金屬凝固過程原子團簇結(jié)構(gòu)表征的新方法

侯兆陽, 劉讓蘇, 田澤安, 王晉國

(1.長安大學(xué)應(yīng)用物理系，西安 710064; 2. 湖南大學(xué)物理與微電子科學(xué)學(xué)院，長沙 410082)

液態(tài)金屬凝固過程原子團簇結(jié)構(gòu)表征的新方法

侯兆陽1, 劉讓蘇2, 田澤安2, 王晉國1

(1.長安大學(xué)應(yīng)用物理系，西安 710064; 2. 湖南大學(xué)物理與微電子科學(xué)學(xué)院，長沙 410082)

采用分子動力學(xué)方法對液態(tài)金屬鈉的凝固過程進行了模擬計算，運用團簇結(jié)構(gòu)表征新方法――團簇類型指數(shù)法(CTIM)對凝固過程中的團簇結(jié)構(gòu)進行了識別.為了闡明CTIM在識別團簇結(jié)構(gòu)上的準確性和效率，將其與廣為采用的Voronoi多面體方法(VPM)進行比較.結(jié)果表明：當采用CTIM和VPM分別對液態(tài)金屬鈉凝固結(jié)構(gòu)中的原子團簇結(jié)構(gòu)進行表征時，它們所得到的微觀結(jié)構(gòu)特征是一致的.非晶態(tài)結(jié)構(gòu)中，原子團簇類型的分布呈現(xiàn)明顯的區(qū)段特征，每一區(qū)段都存在一種主要團簇類型，它們分別是二十面體或其缺陷結(jié)構(gòu).晶體結(jié)構(gòu)中，體系形成以BCC團簇為主體的晶態(tài)結(jié)構(gòu).同時發(fā)現(xiàn)，VPM不易區(qū)分不同團簇構(gòu)型之間的細微差別，不同構(gòu)型的原子團簇可能被歸為同種結(jié)構(gòu)類型；而CTIM根據(jù)近鄰原子之間相對位置關(guān)系，直接準確描述原子團簇的構(gòu)型.不但由CTIM分析獲得的凝固體系結(jié)構(gòu)特征與VPM的分析結(jié)果一致；而且CTIM抓住了體系微觀結(jié)構(gòu)特征的主要方面，簡化了團簇結(jié)構(gòu)的表征形式，這在大尺度模擬體系的結(jié)構(gòu)分析中將具有較高效率.

原子團簇；結(jié)構(gòu)表征方法；分子動力學(xué)；凝固結(jié)構(gòu)

1 Introduction

Atomic clusters are basic configurations in liquid metals, and their evolution properties play an important role in understanding the solidification mechanism[1,2]. At present, computer simulation methods have been widely used to investigate the microstructures during the solidification processes of liquids. However, only atom positions are recorded in the computer simulations, so the method of characterizing atomic clusters is of essential importance to study the solidification mechanism of liquid metals.

Several methods have been proposed in the past to characterize atomic clusters in the solidification processes of liquids. Based on bond-orientational order parameters[3], ten Woldeetal.[4]proposed a method to identify individual atoms as either solid- or liquid-like, and then extended to solid- or liquid-like clusters. This method has gained increasing interests for its simple algorithm and high computational efficiency[5-7], but the detailed topology information of atomic clusters is absent, since the bond-orientational order parameters are statistical average quantities. There are alternative approaches to characterize atomic clusters based on the operation of graphs. One is the Voronoi polyhedron method (VPM), which was first used by Finney as early as 1970[8]. In this method, the configuration of an atomic cluster composed of a central atom along with its surrounding neighbor atoms is characterized by the shape of Voronoi polyhedron associated with the central atom. However, some vertices in the Voronoi polyhedron often split into small faces because of thermal fluctuations and rounding errors in computation, then the Voronoi polyhedron is distorted. Some approximate approaches[9-11]were adopted to eliminate the small faces, but these approximate treatments are not very reasonable since the small faces resulted from thermal vibrations may be comparable to the original ones. In addition, VPM is hard to describe the atomic clusters with larger size as the computational complexity. Another widely used method for cluster characterization is the Honeycutt and Andersen (HA) bond-type index method[12]. This method adopts a set of indices to describe the configuration of atomic cluster composed of a pair of atoms along with their common neighbor atoms. While the atomic clusters described by the HA bond-type index method usually contain no more than ten atoms[13]. In order to characterize the atomic clusters with larger size, we have proposed the cluster-type index method (CTIM)[14-16]based on the HA bond-type index method.

The reliability and efficiency of structural characterization method of atomic clusters are believed to be a key issue to achieve reliable results in the investigation of solidification mechanism of liquids by computer simulations. Accordingly, in this paper, we present the cluster characterization method of CTIM[14-16]proposed by us in detail, and evaluate its reliability and efficiency by comparing it with the widely used VPM. To compare the structural features of atomic clusters obtained by the VPM and CTIM, respectively, the solidification processes of liquid metal Na are simulated since extensive researches have been carried out on it[17-19].

2 Molecular dynamics simulation

The solidification processes of liquid metal Na are simulated by the molecular dynamics (MD) method. MD simulations are performed for a system containing 1000 Na atoms in a cubic box with periodic boundary conditions under constant-pressure. The equations of motion are integrated by the leap-frog algorithm with a time step of 2 fs. The interatomic potential adopted here is the effective pair potential derived from the generalized non-local model pseudopotential (GNMP) based upon the first-principle interaction force in the second order perturbation theory[20, 21]. For simple metals and their alloys, the accuracy and reliability of this effective pair potential have been demonstrated extensively by computing their structural, dynamic and thermodynamics properties[20-23]. The pair potential is cut off at 20 a.u. (atomic unit).

Simulation calculations are started at 973 K (the melting point Tmof Na is near 371 K). First of all, let the system run 20000 time steps at 973 K to obtain the equilibrium liquid determined by the energy changes of system. Then the Gaussian thermostatis adopted to decrease the system temperature to 73 K at two cooling rates of 1×1014K/s and 1×1011K/s. The intervals between two temperature points are 50 K. At each given temperature, the instantaneous spatial coordinates of each atom are recorded for the structural analyses below. Finally, the structural analyses are performed in terms of the radical distribution function (RDF), VPM, and CTIM.

3 Cluster characterization methods

3.1 VPM

The Voronoi polyhedron associated with a given atom is defined as the smallest closed convex polyhedron consisting of the planes that perpendicularly bisect the coordination vectors from the atom to its neighbors[8]. The Voronoi polyhedron corresponds to the Wigner-Seitz cell in the crystalline state. It is customary to define the signature of a Voronoi polyhedron as a set of integers (n3n4n5…ni……), whereniis the number ofi-edged faces of the polyhedron. For example, (0 0 12 0) denotes a polyhedron composed of 12 pentagons, whose shape corresponds to the icosahedron cluster; while the atoms in bcc and fcc environments are associated with the (0 6 0 8) and (0 12 0 0) Voronoi polyhedra, respectively. Their schematic configurations under perfect conditions are shown in Fig. 1. As the thermal fluctuations in the realistic solidification system, Voronoi polyhedra would be distorted, as shown in Fig. 2.

Fig. 1 Schematic configurations of (0 0 12 0), (0 6 0 8) and (0 12 0 0) Voronoi polyhedra under perfect conditions. The quadrangle, pentagon and hexagon faces are colored with cyan, white, and buff, respectively

Fig. 2 Schematic configurations of (0 0 12 0) and (0 6 0 8) Voronoi polyhedra in present simulation system.The quadrangle, pentagon and hexagon faces are colored with cyan, white, and buff, respectively

3.2 CTIM

The HA bond-type index method[12]adopts a set of four integersijklto describe the configuration of an atomic cluster composed of a pair of atoms along with their common neighbor atoms. The first integeriis to identify the bonding of two given atoms.iis 1 when they are bonded in the root pair, otherwise 2. The second integerjis the number of near-neighbor atoms shared in common by the root pair. The third integerkis the number of bonds among the shared neighbors. The fourth integerlis needed to distinguish configurations having the same first three indices but being different bond geometries. In order to characterize atomic clusters with larger size, we have proposed the CTIM[14-16]based on the HA bond-type index. We define the basic cluster as the smallest cluster composed of one central atom along with its nearest-neighbor atoms, and the CTIM adopts four indices (N,n1,n2,n3) to denote different types of basic clusters, whereNis the number of the nearest-neighbor atoms, namely, the coordination number (CN), andn1,n2,n3denotes the numbers of 1441, 1551 and 1661 bond-types, respectively, by these bond-types the surrounding atoms are connected with the central one of the basic cluster. For example, the (12 0 12 0) denotes an icosahedron basic cluster that is composed of one central atom and 12 neighbor atoms (all of them form 1551 bond-type with the central atom); the (14 0 12 2) expresses a Frank-Kasper polyhedron basic cluster that is composed of one central atom and 14 neighbor atoms (twelve of them form 1551 bond-types with the central atom and two of them form 1661 bond-type with the central atom); the (14 6 0 8) stands for a bcc basic cluster (bcc crystal unit) composed of one central atom and 14 near neighbor atoms (six of them form 1441 bond-type with the central atom and eight of them form 1661 bond-type with the central atom), and so on. The schematics of these basic clusters are shown in Fig. 3.

By means of CTIM, many kinds of basic clusters in liquid, amorphous as well as the bcc crystal can be represented effectively[14-15]. However, the familiar fcc crystal unit which is made up of twelve 1421 bond-types, and the hcp crystal unit which is made up of six 1421 and six 1422 bond-types, cannot be described clearly. In order to comprehensively describe crystal clusters (containing hcp and fcc basic clusters), two indices (namely, the fifth and sixth ones which respectively represent the numbers of 1421 and 1422 bond-types) are added to the CTIM. For convenience of discussion, the CTIM with six integers is called as CTIM-2. In CTIM-2, the icosahedron, bcc, fcc and hcp crystal units can be expressed in turn by (12 0 12 0 0 0), (14 6 0 8 0 0), (12 0 0 0 12 0) and (12 0 0 0 6 6), respectively. In the solidification processes of liquid Na, fcc and hcp basic clusters are very scarce, so we adopt the CTIM with four indices in this work for simplification.

Fig. 3 Schematic configurations of (12 0 12 0), (14 0 12 2) and (14 6 0 8) basic clusters in present simulation system. The cyan, white, and yellow spheres denote 1441, 1551, and 1661 bond-types, respectively

Based on the CTIM, extended clusters with large size can be described clearly, for details see Refs. [14,15]. As examples, Fig. 4(a) is a large cluster including 25 atoms combined by three basic clusters; while Fig. 4(b) contains 49 atoms combined by seven basic clusters.

Fig. 4 Schematic configurations of two extended clusters. (Schematic configurations of central atoms at bottom right corner) (a) A large cluster including 25 atoms combined by three different basic clusters [one (13 3 6 4), one (14 1 10 3) and one (14 3 6 5) ]; (b) A large cluster including 49 atoms combined by seven basic clusters [two (13 3 6 4), one (13 4 4 5), two (13 5 2 6) , one (15 4 4 7) and one (15 5 2 8)]. The black spheres are the central atoms, and the white spheres are their surface atoms

4 Results

To verify the validity of our simulation methods, the RDF obtained in present simulations is compared with that of experimental results of Waseda[24]as shown in Fig. 5. It can be seen that the simulation RDF of liquid Na (573 K) are in agreement with the experimental results. This indicates that present simulations are rather successful in describing the physical nature of the system. Moreover, from the RDFs of the solidification solids (73 K) at different cooling rates (see Fig. 5), it can be found that the solidification solid with the cooling rate of 1×1014K/s displays amorphous feature, while distinct crystal characteristics are shown for that with the cooling rate of 1×1011K/s.

Fig. 5 RDFs of liquid Na (573 K), its amorphous and crystal solids (73 K) at cooling rates of 1×1014 K/s and 1×1011 K/s, respectively. The experimental points at 573 K are taken from Ref.[24]

To evaluate the feasibility of CTIM, atomic clusters in the same solidification solids are characterized by the VPM and CTIM, respectively. According to the VPM, the statistical numbers of various Voronoi polyhedra in the amorphous and crystal solids (73 K) can be obtained. A total of 134 different types of Voronoi polyhedra are detected in the amorphous structure, while only 24 types in the crystal structure. Among these Voronoi polyhedra, those only composed of quadrangle, pentagon and hexagon faces are dominated. Their total numbers amount to 87.7% of all Voronoi polyhedra in the amorphous solid, while 96.2% in the crystal solid. Thus we only show the statistical numbers of the Voronoi polyhedra composed of quadrangle, pentagon and hexagon faces in Fig. 6 for clearness. From Fig. 6(a), it can be found that when the different Voronoi polyhedra are arranged in turn according to the numbers of quadrangle, pentagon and hexagon faces, distributions of these polyhedra show six broad peaks at (0 0 12 0), (0 1 10 2), (0 2 8 2), (0 3 6 4), (0 4 4 6) and (0 5 2 6). The (0 0 12 0) Voronoi polyhedron corresponding to icosahedron cluster and its defective cases[25]with the (0 1 10 2), (0 2 8 2), (0 2 8 4), (0 3 6 4) signatures are favorable in the amorphous structure, and their total number reaches to 52.9% of all Voronoi polyhedra in the system. It should be noted that the fraction of (0 0 12 0) is not highest among them, even though it is often referred to as the typical configuration of amorphous structures[26,27]. The fraction of (0 6 0 8) signature which corresponds to the bcc crystal unit is very small, but its defective cases with (0 4 4 6) and (0 5 2 6) signatures[17]are abundant. From Fig. 6(b), it can be found that (0 6 0 8) Voronoi polyhedron is the characteristic cluster of crystal solid, and its fraction reaches 93.6% in the system.

When the atomic clusters in amorphous and crystal solids (73 K) are characterized by the CTIM, the statistical numbers of various basic clusters are obtained, respectively, as shown in Fig. 7. From Fig. 7(a), it can be seen that when the different basis clusters are arranged in turn according to the numbers of 1441, 1551 and 1661 bond-types, distributions of these basic clusters show five broad peaks at (12 0 12 0), (13 1 10 2), (14 2 8 4), (13 3 6 4) and (14 4 4 6). The icosahedron basic cluster (12 0 12 0) and its defective cases (13 1 10 2), (14 1 10 3), (14 2 8 4), (13 3 6 4)[15]are favorable, and their total number occupies 54.4% of all basic clusters in the system. From Fig. 7(b), it can be found that the (14 6 0 8) basic cluster is the characteristic clusters of crystal solid, and its fraction reaches 98.8% in the system.

Fig. 7 Number distributions of basic clusters in the solidification solids (73 K) of liquid Na. (a) Amorphous solid with the cooling rate of 1×1014 K/s, (b) Crystal solid with the cooling rate of 1×1011 K/s

5 Discussion

When we compare the structural features of atomic clusters in the same solidification solids obtained by the VPM and CTIM, respectively, it can be found that their results are consistent with each other. The distributions of atomic clusters in the amorphous structure characterized by the two different methods both display several broad peaks, and each peak corresponds to one favorable cluster type. The favorable cluster types both are icosahedron and its defective cases, described as the (0 0 12 0), (0 1 10 2), (0 2 8 2), (0 2 8 4) and (0 3 6 4) polyhedra in VPM, while the (12 0 12 0), (13 1 10 2), (14 2 8 4) and (13 3 6 4) basic clusters in CTIM. The dominated cluster type in the crystal structure characterized by the two different methods both is bcc crystal unit, described as the (0 6 0 8) polyhedra in VPM, while the (14 6 0 8) basic cluster in CTIM. This indicates the feasibility of CTIM.

The favorable cluster types in solidification structures have similar representation signatures characterized by the VPM and CTIM. To make clear their relationships in topology, the configurations of the same atomic clusters characterized by the two methods are shown together in Fig. 8. It can be found that the VPM indirectly describes the cluster configurations by means of the shapes of Voronoi polyhedra associated with central atoms; while the CTIM directly describes the cluster configurations according to the position relations of neighbor atoms.

Fig.8 Schematic configurations of atomic clusters characterized by VPM and CTIM together. (a) Icosahedron with central atom 327-numbered; (b) bcc crystal unit with central atom 200-numbered. The quadrangle, pentagon and hexagon faces of Voronoi polyhedra are colored with cyan, white, and buff, respectively; while these colors respectively denote 1551, 1441, and 1661 bond-types in CTIM

The neighbor atoms which form 1441, 1551, and 1661 bond-types with central atoms, respectively, correspond to the quadrangle, pentagon and hexagon faces of Voronoi polyhedra. But the inverse is not always true. Namely, the quadrangle, pentagon and hexagon faces of Voronoi polyhedra do not always correspond to the 1441, 1551 and 1661 bond-types, respectively. For example, as shown in Fig. 9, the Voronoi polyhedron associated with the central atom 14-numbered has the (0 0 12 0) signature, reflecting the icosahedron local configuration. However, the neighbor atoms labeled 629 and 879 both form 1431 bond-types with the central atom and those labeled 402 and 879 both form 1543 bond-types, since the distances between 629- and 879-labeled atoms, between 402- and 879 -labeled atoms both are a little farther than the bonding length. Thus this atomic cluster constitutes eight 1551 bond-types, two 1543 bond-types, and two 1431 bond-types. It is not a canonical icosahedron characterized by CTIM. This means that the VPM is difficult to distinguish the small differences between cluster configurations just according to the statistical number of multi-edged faces, and different cluster configurations may be classified into the same type. While the CTIM directly describe the cluster configurations and can exactly present the relative position relations of neighbor atoms.

Fig. 9 Structural configuration of an atomic cluster with central atom 14- numbered characterized by VPM and CTIM together

In VPM, each atom in a system would correspond to a Voronoi polyhedron, and a complete set of these polyhedra form a Voronoi diagram. But the number of basic clusters characterized by CTIM in the amorphous sample is just 167, because not all cluster configurations meet its bonding conditions. It is interesting that though the CTIM only detects a few atomic clusters in the system, the structural features obtained by the two methods are the same. By further analyzing the cluster structures detected by CTIM, we find they are only part of the cluster configurations detected by VPM, in which the faces of Voronoi polyhedra are close to equilateral polygon. This indicates that though the CTIM can not detect all atomic clusters in a system, it still can exactly reflect the feature of microstructures. And it simplifies the representation format of atomic clusters by outstanding the principal ones. This would be efficient for larger-scale systems[28].

Based on the CTIM, all atomic clusters around each atom in a system can also be detected by adjusting the bond-type in the indices (N,n1,n2,n3, …ni…)[29,30]. For example, all crystal phases of Mg-Zn alloy can be characterized by adding 1541, 1321, and 1431 bond-types to the CTIM-2, and distributions of the different phases during diffusion processes can be further detected.

6 Conclusions

An alternative method of CTIM for characterizing atomic clusters in the solidification processes of liquids is proposed by us. The feasibility of CTIM is clarified by comparing it with the widely used VPM.

Our results show that when the atomic clusters in the solidification structures of liquid Na are characterized by the VPM and CTIM, respectively, their structural features are identical. The distributions of atomic clusters in the amorphous structure characterized by the two different methods both display several broad peaks, and each peak corresponds to one favorable cluster type. The favorable cluster types both are icosahedron and its defective cases, described as the (0 0 12 0), (0 1 10 2), (0 2 8 2), (0 2 8 4) and (0 3 6 4) polyhedra in VPM, while the (12 0 12 0), (13 1 10 2), (14 2 8 4) and (13 3 6 4) basic clusters in CTIM. The dominated cluster type in the crystal structure characterized by the two different methods both is bcc crystal unit, described as the (0 6 0 8) polyhedra in VPM, while the (14 6 0 8) basic cluster in CTIM.

It is also found that the VPM indirectly describes cluster configurations by means of the shapes of Voronoi polyhedra associated with central atoms. The VPM is difficult to distinguish the small differences between cluster configurations. Atomic clusters with different configurations may be classified into the same type. The CTIM directly describes the cluster configurations according to the position relations of neighbor atoms. Though the CTIM just describes a part of cluster configurations satisfying certain bonding conditions in a system, the structural features obtained by it are consistent with the VPM. CTIM simplifies the representation format of atomic clusters by outstanding the principal ones. This is efficient for larger-scale systems.

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A new method for structural characterization of atomic clusters in solidification processes of liquid metals

HOU Zhao-Yang1, LIU Rang-Su2, TIAN Ze-An2, WANG Jin-Guo1

(1. Department of Applied Physics, Chang'an University, Xi'an 710064, China;2. School of Physics and Microelectronics Science, Hunan University, Changsha 410082, China )

A molecular dynamics simulation has been performed on the solidification process of liquid Na, and the atomic clusters in the solidification process have been identified by means of a new characterization method -- cluster-type index method (CTIM) proposed by us. In order to evaluate the reliability and efficiency of CTIM, it is compared with the widely used Voronoi polyhedron method (VPM). Our results show that when the atomic clusters in the solidification structures of liquid Na are characterized by the VPM and CTIM, respectively, their structural features are identical. The distributions of atomic clusters in the amorphous structure characterized by the two different methods both display several broad peaks, and each peak corresponds to one favorable cluster type. The favorable cluster types both are icosahedron and its defective cases. The dominated cluster type in the crystal structure characterized by the two different methods both is bcc crystal unit. It is also found that the VPM is difficult to distinguish the small differences between cluster configurations according to the shapes of Voronoi polyhedra, and different cluster configurations may be classified into the same type. The CTIM directly describes cluster configurations according to the position relations of neighbor atoms. CTIM simplifies the representation format of atomic clusters by means of outstanding the principal ones. This will be efficient in the structural analysis of larger-scale simulation systems.

Atomic cluster; Microstructure characterization method; Molecular dynamic simulation; Solidification

103969/j.issn.1000-0364.2015.02.010

2014-05-08

國家自然科學(xué)基金(51101022, 50831003); 中央高?；究蒲袠I(yè)務(wù)費(CHD2012JC096)

侯兆陽(1980—), 河南南陽人，副教授，博士，研究方向為液態(tài)金屬凝固理論.E-mail: zhaoyanghou@163.com

O552.6

A

1000-0364(2015)02-0232-09