**4.2 Evolution of nana-clusters**

In order to display clearly the evolution characteristics of nano-clusters, it is necessary to trace the evolution processes of nano-clusters during rapid solidification processes. Adopting an inverse-evolving method, some tracking studies for the structural configurations of the nano-clusters have been performed. The evolution processes of the nano-clusters, at different temperatures, have been shown in Figures. It can be clearly seen that the central atoms of basic clusters of the nano-clusters are bonded with each other, some central atoms are multi-bonded, and others single-bonded.

In this simulation, some nano-clusters have been found. They are composed of various kinds of smaller clusters, and their size and amount are increased with temperature decreasing. Their configurations are very complex. For example, a nano-cluster consisting of 126 atoms are composed of 24 basic clusters with center atoms (represented by gray circle), as shown in Fig.8 (a), (b). It can be seen that the nano-cluster is produced by combining three different middle clusters, and each middle cluster composed of some basic clusters, and each basic cluster described by a set of indexes in CTIM.

Fig. 7**.** Schematic diagram of a larger cluster consisting of 68 atoms within ten basic clusters with connecting bonds at 350 K (the gray spheres are the center atoms of basic clusters). The cluster is composed of 1 icosahedron (12 0 12 0), and basic clusters of 1 (16 0 12 4), 5 (13 1 10 2), 1 (14 1 10 3), 1 (14 2 8 4) and 1 (14 3 6 5). (a) displays all the atoms; (b) displays only the central atoms.

In order to display clearly the evolution characteristics of nano-clusters, it is necessary to trace the evolution processes of nano-clusters during rapid solidification processes. From our previous simulation results (Liu R S, et al., 1995, 2002), we have known that once an atom became the center of a cluster, it would possess certainly relative stability and

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

In order to display clearly the evolution characteristics of nano-clusters, it is necessary to trace the evolution processes of nano-clusters during rapid solidification processes. Adopting an inverse-evolving method, some tracking studies for the structural configurations of the nano-clusters have been performed. The evolution processes of the nano-clusters, at different temperatures, have been shown in Figures. It can be clearly seen that the central atoms of basic clusters of the nano-clusters are bonded with each other, some

In this simulation, some nano-clusters have been found. They are composed of various kinds of smaller clusters, and their size and amount are increased with temperature decreasing. Their configurations are very complex. For example, a nano-cluster consisting of 126 atoms are composed of 24 basic clusters with center atoms (represented by gray circle), as shown in Fig.8 (a), (b). It can be seen that the nano-cluster is produced by combining three different middle clusters, and each middle cluster composed of some basic clusters,

Fig. 7**.** Schematic diagram of a larger cluster consisting of 68 atoms within ten basic clusters with connecting bonds at 350 K (the gray spheres are the center atoms of basic clusters). The cluster is composed of 1 icosahedron (12 0 12 0), and basic clusters of 1 (16 0 12 4), 5 (13 1 10 2), 1 (14 1 10 3), 1 (14 2 8 4) and 1 (14 3 6 5). (a) displays all the atoms; (b) displays

In order to display clearly the evolution characteristics of nano-clusters, it is necessary to trace the evolution processes of nano-clusters during rapid solidification processes. From our previous simulation results (Liu R S, et al., 1995, 2002), we have known that once an atom became the center of a cluster, it would possess certainly relative stability and

system have been distributed homogeneously.

central atoms are multi-bonded, and others single-bonded.

and each basic cluster described by a set of indexes in CTIM.

**4.2 Evolution of nana-clusters** 

only the central atoms.

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 center (Joshi et al., 2006).

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

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

> Cluster size

Number

Cluster consisting of 7 basic clusters

> Cluster number

of atom 943K350K Number

Cluster consisting of 8 basic clusters

35 0 2 37 0 1 44 0 1 46 0 1 56 0 2 36 0 3 38 0 0 45 0 0 47 0 0 57 0 1 37 0 7 39 0 0 46 0 1 48 0 0 58 0 1 38 1 7 40 0 0 47 0 2 49 0 2 59 0 1 39 2 16 41 0 4 48 0 4 50 0 0 60 0 2 40 0 34 42 1 3 49 0 3 51 0 0 61 0 1 41 1 36 43 1 8 50 1 8 52 0 3 **62** 0 **7 42** 2 **54** 44 0 13 51 0 12 53 0 0 63 0 5 43 0 51 45 0 12 52 0 11 54 0 4 64 0 4 44 2 33 46 0 19 53 0 13 55 0 6 65 0 3 45 0 51 47 0 23 54 0 16 56 0 3 66 0 4 46 0 45 **48** 2 **33** 55 0 17 57 0 7 **67** 0 **9** 47 0 21 49 1 17 56 0 16 58 0 5 68 0 6 48 0 27 50 0 27 57 0 10 59 0 5 69 0 6 49 0 12 51 0 29 58 0 13 60 0 7 70 0 2 50 0 9 52 0 20 **59** 0 **22** 61 0 9 71 0 0 51 0 8 53 0 18 60 0 11 62 0 3 72 0 1 52 0 1 54 0 12 61 0 6 63 0 7 73 0 1 53 0 1 55 0 6 62 0 3 64 0 2 74 0 2 56 1 4 63 0 0 **65** 0 **14** 75 0 1 57 0 4 64 0 1 66 0 4 76 0 0 58 0 3 65 0 1 67 0 1 77 0 0 59 0 1 66 0 1 68 0 6 78 0 1

Cluster number

of atom 943K350K Number

Cluster size

 69 0 1 70 0 2

**4.3.1 Magic number sequence of nana-clusters for liquid metal Al** 

size (number of atoms included) for liquid metal Al.

Table 5. Relations of the number of clusters consisting of 1-10 basic clusters with the cluster

For liquid metal Al, for simplicity, we only analyze ten groups in the system in turn by the numbers of basic clusters contained in each group for two cases of liquid state at 943K and solid state at 350K, as shown in Table 5. From Table 5, it can be clearly seen that there is a peak value (maximum) of the numbers of clusters for each group and this is shown with a short underline in the table. As we compare this peak value with the abundance usually used in the research of cluster configurations, it is found that the two concepts are

Cluster consisting of 6 basic clusters

> Cluster number

> > 943K 350K

Cluster size

Number of atom Liquid Metal Atoms) 187

Cluster number

of atom 943K350K Number

Cluster consisting of 10 basicclusters

of atom 943K350K

Cluster number

Cluster size

Cluster consisting of 9 basic clusters

Cluster size

#### **4.3 Size distribution and magic number sequence of nana-clusters**

In order to investigate the size distribution characteristics of various clusters in the system, the relationship between the numbers of various clusters and their sizes (the numbers of atoms contained in each cluster) should be displayed clearly according to some statistical method. For convenience of discussion, we propose a new statistical method as follows.

Since a larger cluster can be described clearly by different basic clusters in the CTIM, all the clusters (from basic cluster to larger cluster) in the system can be classified according to the numbers of basic clusters contained in the larger cluster under consideration. Then, the clusters containing the same numbers of basic clusters can be further classified as a group. However, the clusters within a same group may not have the same number of atoms because the different basic clusters they contained would have different number of atoms. Thus there is a certain range of the numbers of atoms for a group of clusters, this can be clearly seen below.


(continued)

In order to investigate the size distribution characteristics of various clusters in the system, the relationship between the numbers of various clusters and their sizes (the numbers of atoms contained in each cluster) should be displayed clearly according to some statistical method. For convenience of discussion, we propose a new statistical method as follows.

Since a larger cluster can be described clearly by different basic clusters in the CTIM, all the clusters (from basic cluster to larger cluster) in the system can be classified according to the numbers of basic clusters contained in the larger cluster under consideration. Then, the clusters containing the same numbers of basic clusters can be further classified as a group. However, the clusters within a same group may not have the same number of atoms because the different basic clusters they contained would have different number of atoms. Thus there is a certain range of the numbers of atoms for a group of clusters, this can be

> Cluster consisting of 3 basic clusters

> > Cluster number

of atom 943K350K Number

**19** 167 **2761 25** 28 **647** 28 2 47 32 0 9

**20 269** 1432 26 53 551 29 4 130 33 1 23

21 311 1370 **27 62 588** 30 10 128 34 3 46

11 13 0 17 1 0 23 9 210 26 0 7 30 0 2 12 245 55 18 6 4 24 19 204 27 0 27 31 0 6

16 300 433 22 202 730 28 43 451 31 8 212 35 0 69 17 39 37 23 86 273 29 32 202 32 9 210 36 4 69 18 0 1 24 46 56 30 21 106 **33** 10 **225** 37 1 105 25 13 15 31 3 38 34 7 194 **38 7 118**  26 2 1 32 4 15 **35 15** 126 39 6 106 33 1 3 36 3 83 40 0 110 37 4 27 41 2 69 38 1 17 42 1 46 39 1 7 43 1 27 40 0 1 44 1 15 41 0 1 45 0 7 42 0 2 46 0 5 47 0 2

Cluster size

Cluster consisting of 4 basic clusters

> Cluster number

of atom 943K350K Number

Cluster size

Cluster consisting of 5 basic clusters

of atom 943K350K

Cluster number

Cluster size

**4.3 Size distribution and magic number sequence of nana-clusters** 

Cluster consisting of 2 basic clusters

> Cluster number

of atom 943K350K Number

Cluster size

Number

clearly seen below.

Cluster consisting of 1 basic clusters

> Cluster number

> > 943K 350K

**2254 10606** 

1912 3159

998 1611

Cluster size

Number of atom

**13** 

14

15

(continued)


Table 5. Relations of the number of clusters consisting of 1-10 basic clusters with the cluster size (number of atoms included) for liquid metal Al.
