**3.3 Cluster-type index analysis**

As is well known that different combinations of bond-types can form different cluster configurations, however, the HA bond-type indices cannot be used to describe and discern different basic clusters formed by an atom with its nearest neighbors, especially, the different nano-clusters formed by some different basic clusters.

In order to differentiate the basic cluster and the polyhedron, we define the "basic cluster" as the smallest cluster composed of a core atom and its surrounding neighbor atoms. A larger cluster can be formed by continuous expansion, with a basic cluster as the core, according to a certain rule, or by combining several basic clusters together. A polyhedron is generally a hollow structure with no central atom as the core. This is the essential distinction of a polyhedron from a "basic cluster", such as the Bernal polyhedron. However, if a basic cluster is shaped as a certain polyhedron, for simplicity, we also call it a polyhedron cluster, such as icosahedral cluster, Bernal polyhedron cluster, and so on.

It is clear that the bond-types formed by each atom with its neighbor atoms in the system are different; the cluster configurations formed by these bond-types are also different. Even if some cluster configurations are formed by the same number of bond-types, their structures may still be completely different from each other, owing to a slight difference in bondlength or bond-angle. On this point, at present it is hard to use the bond-type index method

Formation and Evolution Characteristics of Nano-Clusters (For Large-Scale Systems of 106

Table 3**.** Variation of the number of clusters with temperature (K) for liquid metal Al.

Liquid Metal Atoms) 179

to describe clearly the cluster configurations of different types. In order to deal with this difficult matter, a cluster-type index method (CTIM) has been proposed (Liu R. S., et al., 1998, 1999, 2005a, 2005b; Dong K. J., et al., 2003) based on the HA indexes (Honeycutt & Andersen, 1987) and the work of Qi and Wang (Qi. & Wang, S., 1991b). According to the definition of basic cluster, four integers (N, n1441,n1551,n1661) also adopted to describe the basic clusters. The meaning of the four integers used in CTIM are as follows: the first integer represents the number of surrounding atoms which form a basic cluster with the central atom, i. e. the coordination number Z of the central atom; the second, third and fourth integers respectively represent the numbers of 1441, 1551 and 1661 bond-types, by which the surrounding atoms are connected with the central atom of the basic cluster. For example, (12 0 12 0) stands for an icosahedral cluster that is composed of 13 atoms: the central atom is connected to the surrounding atoms through twelve 1551 bond-types (i. e. the coordination number of the central atom Z=12); (13 1 10 2) stands for the defective polyhedron cluster composed of 14 atoms, the core atom is connected to the surrounding atoms with one 1441, ten 1551 and two 1661 bond-types (the coordination number Z=13). For ease of representation, some main basic clusters have been chosen from the simulation system of liquid metal Al as shown in Fig. 3.

By using the CTIM, the statistical numbers of various cluster-types at each given temperature have been obtained. For liquid metal Al, during the whole solidification process, there are 53 different basic cluster-types in the system, only 34 of them appearing more than 5 times at some temperatures are listed in Table 3. For liquid metal Na, there are 63 different basic cluster-types, of which only 34 main types are listed in Table 4 ( only 17 types appearing more than 1000 times play a critical role).

Fig. 3. Schematics of five main basic clusters at 350K: (a) icosahedral cluster(12 0 12 0) with central atom of 238228; (b) basic cluster(13 1 10 2)with central atom of 354408; (c) basic cluster(14 2 8 4)with central atom of 205450; (d) basic cluster(14 1 10 3)with central atom of 652129; (e) basic cluster(12 2 8 2)with central atom of 778825.

to describe clearly the cluster configurations of different types. In order to deal with this difficult matter, a cluster-type index method (CTIM) has been proposed (Liu R. S., et al., 1998, 1999, 2005a, 2005b; Dong K. J., et al., 2003) based on the HA indexes (Honeycutt & Andersen, 1987) and the work of Qi and Wang (Qi. & Wang, S., 1991b). According to the definition of basic cluster, four integers (N, n1441,n1551,n1661) also adopted to describe the basic clusters. The meaning of the four integers used in CTIM are as follows: the first integer represents the number of surrounding atoms which form a basic cluster with the central atom, i. e. the coordination number Z of the central atom; the second, third and fourth integers respectively represent the numbers of 1441, 1551 and 1661 bond-types, by which the surrounding atoms are connected with the central atom of the basic cluster. For example, (12 0 12 0) stands for an icosahedral cluster that is composed of 13 atoms: the central atom is connected to the surrounding atoms through twelve 1551 bond-types (i. e. the coordination number of the central atom Z=12); (13 1 10 2) stands for the defective polyhedron cluster composed of 14 atoms, the core atom is connected to the surrounding atoms with one 1441, ten 1551 and two 1661 bond-types (the coordination number Z=13). For ease of representation, some main basic clusters have been chosen from the simulation system of

By using the CTIM, the statistical numbers of various cluster-types at each given temperature have been obtained. For liquid metal Al, during the whole solidification process, there are 53 different basic cluster-types in the system, only 34 of them appearing more than 5 times at some temperatures are listed in Table 3. For liquid metal Na, there are 63 different basic cluster-types, of which only 34 main types are listed in Table 4 ( only 17

Fig. 3. Schematics of five main basic clusters at 350K: (a) icosahedral cluster(12 0 12 0) with central atom of 238228; (b) basic cluster(13 1 10 2)with central atom of 354408; (c) basic cluster(14 2 8 4)with central atom of 205450; (d) basic cluster(14 1 10 3)with central atom of 652129; (e) basic cluster(12 2 8 2)with central atom of 778825.

liquid metal Al as shown in Fig. 3.

types appearing more than 1000 times play a critical role).


Table 3**.** Variation of the number of clusters with temperature (K) for liquid metal Al.



Formation and Evolution Characteristics of Nano-Clusters (For Large-Scale Systems of 106

plays a certain role.

secondary role.

temperature Tg.

Number of basic clusters in system

Liquid Metal Atoms) 181

T / K 300 400 500 600 700 800 900 1000

(13 3 6 4) (14 0 12 2) (15 1 10 4) (13 2 8 3) (15 2 8 5)

For liquid metal Al, from Table 3, however, it can be clearly seen that only 18 cluster-types appearing more than 100 times play a critical role in the solidification process. For convenience of discussion, we only show the variations of ten significant basic clusters with temperature in Figure 4(a) and (b). From Figure 4 (a), it is clear that among the most significant five basic clusters, the highest one is the icosahedral basic cluster expressed by (12 0 12 0), which increases rapidly as the temperature comes down. The number of basic cluster (12 0 12 0) climbs above 30,000 at 350K, and this cluster-type plays the most important role in the microstructure transitions during rapid solidification process. The second one is the basic cluster expressed by (13 1 10 2) and its number over 9400 at 350K. The number of the fifth cluster type expressed by (12 2 8 2) is over 1956 at 350K and still

Figure 4 (b) presents that the numbers of basic clusters (14 0 12 2), (15 1 10 4), (13 2 8 3) and (15 2 8 5) all changed at about the same rate, except the basic cluster (13 3 6 4), and their numbers are in the range of 1023 - 1360 at 350K. Therefore, these clusters play only a

However, as we go further to observe the Figures 4 (a), (b) and Table 3 carefully, it can be seen that the basic clusters (12 0 12 0) and (13 3 6 4) have almost a same turning point Tt in the range of 550-625K; in particular for cluster (13 3 6 4), Tt is a peak value point, it means that the cluster (13 3 6 4) plays an opposite role to the cluster (12 0 12 0) in the solidification process. Maybe just these cluster-types play a particular role, this turning point Tt is in agreement with the glass transition temperature Tg obtained by Liu et al ( Liu R. S., Qi & Wang S., 1992; Wendt & Abraham, 1978; Zheng *et al*., 2002). On the other hand, this confirms that the glass transition temperature Tg can also be found by the turning point Tt in the relations of the numbers of main basic clusters with temperature. Therefore, it is possible to find a new method to determine the glass transition

(a) (b)

(12 0 12 0) (13 1 10 2) (14 2 8 4) (14 1 10 3) (12 2 8 2)

T / K 300 400 500 600 700 800 900 1000

Fig. 4. Variation of the numbers of the ten main basic clusters with temperatures for Al.

Number of basic clusters in system

Table 4. Variation of the number of clusters with temperatures (K) for liquid metal Na.

Table 4. Variation of the number of clusters with temperatures (K) for liquid metal Na.

For liquid metal Al, from Table 3, however, it can be clearly seen that only 18 cluster-types appearing more than 100 times play a critical role in the solidification process. For convenience of discussion, we only show the variations of ten significant basic clusters with temperature in Figure 4(a) and (b). From Figure 4 (a), it is clear that among the most significant five basic clusters, the highest one is the icosahedral basic cluster expressed by (12 0 12 0), which increases rapidly as the temperature comes down. The number of basic cluster (12 0 12 0) climbs above 30,000 at 350K, and this cluster-type plays the most important role in the microstructure transitions during rapid solidification process. The second one is the basic cluster expressed by (13 1 10 2) and its number over 9400 at 350K. The number of the fifth cluster type expressed by (12 2 8 2) is over 1956 at 350K and still plays a certain role.

Figure 4 (b) presents that the numbers of basic clusters (14 0 12 2), (15 1 10 4), (13 2 8 3) and (15 2 8 5) all changed at about the same rate, except the basic cluster (13 3 6 4), and their numbers are in the range of 1023 - 1360 at 350K. Therefore, these clusters play only a secondary role.

However, as we go further to observe the Figures 4 (a), (b) and Table 3 carefully, it can be seen that the basic clusters (12 0 12 0) and (13 3 6 4) have almost a same turning point Tt in the range of 550-625K; in particular for cluster (13 3 6 4), Tt is a peak value point, it means that the cluster (13 3 6 4) plays an opposite role to the cluster (12 0 12 0) in the solidification process. Maybe just these cluster-types play a particular role, this turning point Tt is in agreement with the glass transition temperature Tg obtained by Liu et al ( Liu R. S., Qi & Wang S., 1992; Wendt & Abraham, 1978; Zheng *et al*., 2002). On the other hand, this confirms that the glass transition temperature Tg can also be found by the turning point Tt in the relations of the numbers of main basic clusters with temperature. Therefore, it is possible to find a new method to determine the glass transition temperature Tg.

Fig. 4. Variation of the numbers of the ten main basic clusters with temperatures for Al.

Formation and Evolution Characteristics of Nano-Clusters (For Large-Scale Systems of 106

combining several basic clusters together.

different sizes and shapes.

in the near future.

(a) (b)

of (111) cross section×2 times ; (b) a part of (111) cross section ×5 times.

Liquid Metal Atoms) 183

continuous expansion, with a basic cluster as the core, according to a certain rule, or by

Fig. 6. The 2D schematic of the whole system consisting of 1,000,000 atoms at 350K: (a) a part

First of all, we display the whole schematic diagram of the 2D (111) cross section of the 1000000 atoms system of Al at 350 K, as shown in Fig.6(a) and (b), a part of (111) cross section × 2 times and a part of (111) cross section × 5 times, respectively. Fig.6(a), (b) show that the system has become amorphous state and formed two types of region: the dense region and the loose region. In the dense regions, some regular or distorted five-lateral patterns appear in Fig.6(b), which are just the cross sections of some icosahedron and their combining configurations. The loose regions are also of different sizes and shapes without apparent regulation and the atoms are randomly distributed there. The dense regions and the loose regions are also distributed randomly in the system; the inhomogeneous solid seems to be rather sponge-like with cavities (also commonly called "free volume") in

It is clear that the microstructure of this system is hard to be described by the well-known model of "random hard sphere packing", since that model is too simple for describing amorphous metals. Figure 6, however, shows a typical amorphous picture, thus it is necessary to establish a new model to describe the complex structures of amorphous metals

In these simulations, some larger clusters have been found. They are composed of various kinds of basic clusters and their sizes and numbers increase with temperature decreasing. Their configurations are very complex. For example, a larger cluster consisting of 68 atoms is composed of 10 basic clusters with central atoms (represented by gray circle) in Al system as shown in figure 7(a) , (b), displaying the whole atoms and the central atoms, respectively. From figures 7(a) and (b), it can be clearly seen that the larger cluster is formed by combining different medium-sized clusters, and each medium-sized cluster is also composed of some basic clusters that can be described by a set of indexes in the CTIM as shown in the caption. Interestingly, the larger clusters formed during rapid solidification processes of liquid metals Al and Na do not consist of multi-shell configurations

Fig. 5. Relations of the numbers of 10 main basic clusters with temperature for Na.

It is worth noting that, from Table 3, it can be clearly seen that even at 943K, there are still a certain number of various basic clusters in the liquid state. That is to say, the liquid state discussed here is not an ideal liquid, as usually imagined, in which no cluster exist and each atom is free to diffuse. Furthermore, from our previous works for a small system consisting of 500 Al atoms, as shown in Fig.3 of Ref ( Liu R. S., et al, 1999), it can be seen that even the temperature is increased up to 1800K (≈ 2Tm ), the number of 1551 bond-type (which plays a leading role in microstructure transition of liquid metal Al) is still occupied 7.3 % of the total bond-types ( and 16.5 % at 943K); thus some basic clusters formed mainly by 1551 bond-type would still be in the liquid system. If we want to get an ideal liquid state, the temperature should be increased higher and higher. In general, from the view point of microscopic structure, it is hard to completely reach the ideal case.

For liquid metal Na, for convenience of discussion, only the relations of the former 10 main basic clusters with temperature are shown in Fig.5. From Fig.5 (a), it can be clearly seen that the first 3 basic clusters (13 3 6 4), (13 1 10 2) and (14 2 8 4) are increased rapidly with decreasing temperature, and play almost the same important role in the microstructure transitions of liquid metal Na. While the basic cluster (12 0 12 0) has been ranked as the sixth one and only plays a secondary role; however, in the liquid metal Al, it is the first one and plays the most important role in the microstructure transitions ( Liu R. S. et al., 2005a).

#### **4. Formation and evolution of nano-clusters**

#### **4.1 Formation and description of nana-clusters**

In this section, some nano-clusters have been described. They are composed of various kinds of smaller clusters, and their sizes and amounts are increased with temperature decreasing. Their configurations are very complex.

As above mentioned, we have defined the basic cluster as the smallest cluster composed of a core atom and its surrounding neighbor atoms. A larger cluster can be formed by

Number of basic clusters in system

0

2000

4000

6000

8000

(13 3 6 4) (13 1 10 2) (14 2 8 4) (14 3 6 5) (14 4 4 6)

(a) (b)

T / K 200 400 600 800 1000 1200

Number of cluster polyhedron in system

structure, it is hard to completely reach the ideal case.

**4. Formation and evolution of nano-clusters 4.1 Formation and description of nana-clusters** 

decreasing. Their configurations are very complex.

Fig. 5. Relations of the numbers of 10 main basic clusters with temperature for Na.

It is worth noting that, from Table 3, it can be clearly seen that even at 943K, there are still a certain number of various basic clusters in the liquid state. That is to say, the liquid state discussed here is not an ideal liquid, as usually imagined, in which no cluster exist and each atom is free to diffuse. Furthermore, from our previous works for a small system consisting of 500 Al atoms, as shown in Fig.3 of Ref ( Liu R. S., et al, 1999), it can be seen that even the temperature is increased up to 1800K (≈ 2Tm ), the number of 1551 bond-type (which plays a leading role in microstructure transition of liquid metal Al) is still occupied 7.3 % of the total bond-types ( and 16.5 % at 943K); thus some basic clusters formed mainly by 1551 bond-type would still be in the liquid system. If we want to get an ideal liquid state, the temperature should be increased higher and higher. In general, from the view point of microscopic

T / K 200 400 600 800 1000 1200

(12 0 12 0) (14 1 10 3) (12 2 8 2) (15 2 8 5) (15 1 10 4)

For liquid metal Na, for convenience of discussion, only the relations of the former 10 main basic clusters with temperature are shown in Fig.5. From Fig.5 (a), it can be clearly seen that the first 3 basic clusters (13 3 6 4), (13 1 10 2) and (14 2 8 4) are increased rapidly with decreasing temperature, and play almost the same important role in the microstructure transitions of liquid metal Na. While the basic cluster (12 0 12 0) has been ranked as the sixth one and only plays a secondary role; however, in the liquid metal Al, it is the first one and plays the most important role in the microstructure transitions ( Liu R. S. et al., 2005a).

In this section, some nano-clusters have been described. They are composed of various kinds of smaller clusters, and their sizes and amounts are increased with temperature

As above mentioned, we have defined the basic cluster as the smallest cluster composed of a core atom and its surrounding neighbor atoms. A larger cluster can be formed by continuous expansion, with a basic cluster as the core, according to a certain rule, or by combining several basic clusters together.

Fig. 6. The 2D schematic of the whole system consisting of 1,000,000 atoms at 350K: (a) a part of (111) cross section×2 times ; (b) a part of (111) cross section ×5 times.

First of all, we display the whole schematic diagram of the 2D (111) cross section of the 1000000 atoms system of Al at 350 K, as shown in Fig.6(a) and (b), a part of (111) cross section × 2 times and a part of (111) cross section × 5 times, respectively. Fig.6(a), (b) show that the system has become amorphous state and formed two types of region: the dense region and the loose region. In the dense regions, some regular or distorted five-lateral patterns appear in Fig.6(b), which are just the cross sections of some icosahedron and their combining configurations. The loose regions are also of different sizes and shapes without apparent regulation and the atoms are randomly distributed there. The dense regions and the loose regions are also distributed randomly in the system; the inhomogeneous solid seems to be rather sponge-like with cavities (also commonly called "free volume") in different sizes and shapes.

It is clear that the microstructure of this system is hard to be described by the well-known model of "random hard sphere packing", since that model is too simple for describing amorphous metals. Figure 6, however, shows a typical amorphous picture, thus it is necessary to establish a new model to describe the complex structures of amorphous metals in the near future.

In these simulations, some larger clusters have been found. They are composed of various kinds of basic clusters and their sizes and numbers increase with temperature decreasing. Their configurations are very complex. For example, a larger cluster consisting of 68 atoms is composed of 10 basic clusters with central atoms (represented by gray circle) in Al system as shown in figure 7(a) , (b), displaying the whole atoms and the central atoms, respectively. From figures 7(a) and (b), it can be clearly seen that the larger cluster is formed by combining different medium-sized clusters, and each medium-sized cluster is also composed of some basic clusters that can be described by a set of indexes in the CTIM as shown in the caption. Interestingly, the larger clusters formed during rapid solidification processes of liquid metals Al and Na do not consist of multi-shell configurations

Formation and Evolution Characteristics of Nano-Clusters (For Large-Scale Systems of 106

center (Joshi et al., 2006).

Liquid Metal Atoms) 185

continuity (namely heredity). According to this feature, we can adopt the label of the central atom of a basic cluster to simplify the description of the nano-clusters, thus we can understand the whole evolution process of them more clearly. Adopting an inverseevolving method, a tracking study for the structural configurations of this nano-cluster has been made.The evolution process of the nano-cluster, at different temperatures (for simplicity, we only select 2 different temperatures), has been shown in Fig.8(c),(d). It can be clearly seen that when the temperature is below 350K, the central atoms of 24 basic clusters of the nano-cluster are bonded with each other, some central atoms are multi-bonded, and others single-bonded. However, this is a very important characteristic for simplifying the research on the evolution processes and mechanisms of nano-clusters. With the increase of temperature, the maximal size of the original middle and small clusters decreases continuously. From the macro-viewpoint, such a degree of order is rather consistent with the statistical rules of thermodynamics. It can be clearly seen that this nano-cluster is also formed by connecting various middle and small clusters with different cluster-types or sizes, and different from that obtained by gaseous deposition, ionic spray and so on. It is well known that the latter is proved by mass-spectrometric analysis to be the nano-level crystal clusters formed by octahedron-shells configuration accumulated with an atom as the

Fig. 8. Schematic figures of a nano-clusters consisting of 126 atoms within 24 basic clusters with connecting bonds at 350 K(the gray spheres are the center atoms of basic clusters). The cluster is composed of 7 icosahedron (12 0 12 0), and basic clusters of 1 (14 0 12 2), 5 (13 1 10 2), 3 (14 1 10 3), 3 (15 1 10 4), 1 (12 2 8 2), 2 (14 2 8 4), 1 (15 2 8 5) and 1 (15 3 6 6).

(a) (b)

(c) (d)

(a) the whole atoms; (b) at 350K; (c) at 550K; (d) at 780K.

accumulated by atoms as obtained by gaseous deposition or ionic spray methods. However, the cluster configurations of Al formed by gaseous deposition have been verified by massspectrometer to be crystals or similar structures formed in octahedral shell structures (Martin, et al., 1992). Therefore, it can be concluded that different methods of preparing metallic materials would produce different cluster configurations. Figure 7 shows that the atoms contained in the larger clusters are labeled randomly, that is to say, the atoms in the system have been distributed homogeneously.
