**1. Introduction**

It is well known that the formation and evolution characteristics of clusters and nano-clusters have been studied, both experimentally and theoretically over the years. Many experimental works were carried out by using physical or chemical methods, such as ionic spray, thermal evaporation, chemical action deposition, and so on, to obtain some nice particles or clusters consisted of dozens to hundreds of atoms in special configurations ( Echt et al, 1981; Knight et al, 1984; Harris et al, 1984; Schriver et al, 1990; Robles et al, 2002; Magudoapathy et al, 2001; Spiridis et al, 2001; Liu X H et al, 1998; Yamamoto et al, 2001; Bruhl R et al, 2004; Kostko et al, 2007; Alexander & Moshe, 2001) . The theoretical works were mainly carried out on diversified individual clusters configured by accumulating atoms according to some fixed pattern (Liu C. S. et al, 2001; Solov'yov et al, 2003; Doye & Meyer, 2005; Li H. & Pederiva, 2003; Ikeshoji et al, 1996; Wang L et al, 2002; Haberland et al., 2005; Joshi et al., 2006; Noya et al., 2007; Cabarcos et al., 1999; Orlando & James, 1999; Alfe, 2003). However, it is interesting that the similar clusters or aggregations have been found in some liquid metals during rapid solidification processes in our MD simulations ( Liu R. S. et al., 1992a, 1992b, 1995, 2002, 2005a, 2005b, 2007a, 2007b, 2007c, 2009; Dong K. J. et al., 2003; Liu F. X. et al., 2009; Hou Z. Y. et al., 2009, 2010a, 2010b) and that it is also important for understanding in depth the solidification processes from liquid state to solid state. Furthermore, the formation and evolution characteristics of cluster configurations, especially the nano-cluster configurations, formed during solidification processes of liquid metals are still not well known up to now.

In this chapter, the main purpose is to further extended our previous MD simulation method (Liu R. S. et al., 2007a, 2007b, 2007c, 2009; Tian et al., 2008, 2009; Zhou et al., 2011) to study the large-sized systems consisting of 106 atoms of liquid metal Al and Na. Using the center-atom method, bond-type index method, and cluster-type index method (we proposed), the results have been analyzed and demonstrated that the larger simulation system can lead to a better understanding of the formation and evolution characteristics of the cluster configurations, especially the nano-clusters during solidification processes.

<sup>\*</sup> Corresponding Author

#### **2. Simulation conditions and methods**

The molecular dynamics (MD) technique used here is based on canonical MD, and the simulation conditions are as follows: 106 atoms of metal Al, and the same for Na, are placed in a cubic box, respectively, and the systems run under periodic boundary conditions. The cubic box sizes are determined by both the number of atoms in each system and the mean volume of each atom at each given temperature, for these simulations the mean volumes are taken from the Ω-T curve as shown in Fig.5 of Ref (Qi D. W. & Wang S, 1991a) thus the box sizes would be changed with temperature. The motion equations are solved using leap-frog algorithm. The interacting inter-atomic potentials adopted here are the effective pair potential function of the generalized energy independent non-local model-pseudo-potential theory developed by Wang et al (Wang S. & Lai S. K., 1980; Li D. H., Li X. R. & Wang S., 1986). The effective pair potential function is

$$V(r) = \left(Z\_{eff}^2 \;/\; r\right) \left[1 - \left(\frac{2}{\pi}\right) \stackrel{\circ}{\int} dq F(q) \sin(rq) / \; q\right] \tag{1}$$

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

nature of these systems.

**3.2 Bond-type index analysis** 

Liquid Metal Atoms) 175

Figures 1 and 2 for Al and Na, respectively. From these Figures, it can be clearly seen that the simulation results are in good agreement with the experimental results. This means that the effective pair-potentials adopted here have successfully described the objective physical

For deep understanding the formation and evolution mechanism of clusters in liquid metals, it is very important for us to know the concrete relationship of an atom with its near neighbors. Recently, the pair analysis technique has become an important method to describe and discern the concrete relationship of an atom with the near neighbors in liquid and amorphous systems. For a long time, the pair analysis technique, especially, the Honeycutt-Andersen (HA) bond-type indexes (Honeycutt & Andersen, 1987) have been successfully applied to describe and analyze the microstructure transitions in simulation systems, the details was shown in Refs (Liu R. S., et al., 2002; Dong K. J., et al., 2003). In this chapter, for the systems consisting of 106 atoms for Al and Na, various bond-types are also

> r / nm 0.0 .2 .4 .6 .8 1.0 1.2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

r/nm

Theory Experiment

> 573K 473K 373K E573K E473K E378K

described by HA indexes, as shown in Table 1 and 2, respectively.

Fig. 1. Pair distribution function of liquid Al at 943K

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

g(r)

g(r)


0

1

2

3

4

Fig. 2. Pair distribution function of liquid Na at 573K, 473K and 373K

where *Zeff* and *F(q)* are, respectively, the effective ionic valence and the normalized energy wave number characteristics, which were defined in detail in Refs. (Wang S. & Lai S. K., 1980; Li D. H., Li X. R. & Wang S., 1986). These pair potentials are cut off at 20 a.u (atom unit). The time step of simulation is chosen as 10-15s.

The simulating calculations are performed for different metals respectively. For example, the simulation starts at 943K (the melting point (Tm) of Al is 933K), (for other metals at different temperatures). First of all, let the system run at the same temperature so as to reach an equilibrium liquid state determined by the energy change of system. Thereafter, the damped force method (Hoover et al., 1982; Evans, 1983) is employed to decrease the temperature of the system with the cooling rate of 1.00×1013 K/s to some given temperatures: 883, 833, 780, 730, 675, 625, 550, 500, 450, 400 and 350K. At each given temperature, the instantaneous spatial coordinates of each atom are recorded for analysis below. The bond-type index method of Honeycutt-Andersen (HA) ( Honeycutt & Andersen, 1987), the center-atom method ( Liu R. S., Li J. Y. & Zhou Q. Y., 1995 ) and the cluster-type index method (Dong et al., 2003; Liu R. S., et al., 2005a ) are used to detect and analyze the bond-types and cluster-types of the related atoms in the system, and we go further to investigate the formation mechanisms and magic number characteristics of various clusters configurations formed during solidification processes at atomic level as follows.

#### **3. Microstructure analysis**

#### **3.1 Pair distribution function**

The pair distribution function *g*(*r*) can be obtained by Fourier transformation of X-ray scattering structure factors *S*αβ(*Q*), and has been widely used to describe the structure characterization for liquid and amorphous metals. The validity of the simulation results can be verified by comparing the calculated pair distribution function *g*(*r*) with the experimental results.

In this chapter, we inspect the g(r) curves of the system of Al at 943K and of the system of Na at 573K, 473K and 373K obtained from simulations and compare them with the experimental results obtained by Waseda (Waseda, 1980), as shown in corresponding Figures 1 and 2 for Al and Na, respectively. From these Figures, it can be clearly seen that the simulation results are in good agreement with the experimental results. This means that the effective pair-potentials adopted here have successfully described the objective physical nature of these systems.

#### **3.2 Bond-type index analysis**

174 Molecular Dynamics – Theoretical Developments and Applications in Nanotechnology and Energy

The molecular dynamics (MD) technique used here is based on canonical MD, and the simulation conditions are as follows: 106 atoms of metal Al, and the same for Na, are placed in a cubic box, respectively, and the systems run under periodic boundary conditions. The cubic box sizes are determined by both the number of atoms in each system and the mean volume of each atom at each given temperature, for these simulations the mean volumes are taken from the Ω-T curve as shown in Fig.5 of Ref (Qi D. W. & Wang S, 1991a) thus the box sizes would be changed with temperature. The motion equations are solved using leap-frog algorithm. The interacting inter-atomic potentials adopted here are the effective pair potential function of the generalized energy independent non-local model-pseudo-potential theory developed by Wang et al (Wang S. & Lai S. K., 1980; Li D. H., Li X. R. & Wang S.,

<sup>2</sup> <sup>2</sup> *V r Z r dqF q rq q eff* / 1 sin /

where *Zeff* and *F(q)* are, respectively, the effective ionic valence and the normalized energy wave number characteristics, which were defined in detail in Refs. (Wang S. & Lai S. K., 1980; Li D. H., Li X. R. & Wang S., 1986). These pair potentials are cut off at 20 a.u (atom

The simulating calculations are performed for different metals respectively. For example, the simulation starts at 943K (the melting point (Tm) of Al is 933K), (for other metals at different temperatures). First of all, let the system run at the same temperature so as to reach an equilibrium liquid state determined by the energy change of system. Thereafter, the damped force method (Hoover et al., 1982; Evans, 1983) is employed to decrease the temperature of the system with the cooling rate of 1.00×1013 K/s to some given temperatures: 883, 833, 780, 730, 675, 625, 550, 500, 450, 400 and 350K. At each given temperature, the instantaneous spatial coordinates of each atom are recorded for analysis below. The bond-type index method of Honeycutt-Andersen (HA) ( Honeycutt & Andersen, 1987), the center-atom method ( Liu R. S., Li J. Y. & Zhou Q. Y., 1995 ) and the cluster-type index method (Dong et al., 2003; Liu R. S., et al., 2005a ) are used to detect and analyze the bond-types and cluster-types of the related atoms in the system, and we go further to investigate the formation mechanisms and magic number characteristics of various clusters

configurations formed during solidification processes at atomic level as follows.

The pair distribution function *g*(*r*) can be obtained by Fourier transformation of X-ray scattering structure factors *S*αβ(*Q*), and has been widely used to describe the structure characterization for liquid and amorphous metals. The validity of the simulation results can be verified by

In this chapter, we inspect the g(r) curves of the system of Al at 943K and of the system of Na at 573K, 473K and 373K obtained from simulations and compare them with the experimental results obtained by Waseda (Waseda, 1980), as shown in corresponding

comparing the calculated pair distribution function *g*(*r*) with the experimental results.

 

(1)

**2. Simulation conditions and methods** 

1986). The effective pair potential function is

unit). The time step of simulation is chosen as 10-15s.

**3. Microstructure analysis 3.1 Pair distribution function** 

For deep understanding the formation and evolution mechanism of clusters in liquid metals, it is very important for us to know the concrete relationship of an atom with its near neighbors. Recently, the pair analysis technique has become an important method to describe and discern the concrete relationship of an atom with the near neighbors in liquid and amorphous systems. For a long time, the pair analysis technique, especially, the Honeycutt-Andersen (HA) bond-type indexes (Honeycutt & Andersen, 1987) have been successfully applied to describe and analyze the microstructure transitions in simulation systems, the details was shown in Refs (Liu R. S., et al., 2002; Dong K. J., et al., 2003). In this chapter, for the systems consisting of 106 atoms for Al and Na, various bond-types are also described by HA indexes, as shown in Table 1 and 2, respectively.

Fig. 1. Pair distribution function of liquid Al at 943K

Fig. 2. Pair distribution function of liquid Na at 573K, 473K and 373K

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

**3.3 Cluster-type index analysis** 

Liquid Metal Atoms) 177

Table 2. Relations of the number of various bond types (%) of Na with temperature (K).

other, there being only a minor difference during solidification processes.

different nano-clusters formed by some different basic clusters.

such as icosahedral cluster, Bernal polyhedron cluster, and so on.

On the whole, these simulation results are rather close to those obtained in our previous works on different-sized liquid metal systems (Liu R. S., et al., 1998, 1999, 2005a, 2005b; Dong K. J., et al., 2003) ; that is to say, for different-sized liquid metal systems, the simulation results of relative numbers of corresponding bond-types are similar to each

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

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,

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

From Table 1 and 2, it can be clearly seen that:

Firstly, the relative numbers of 1551 and 1541 bond-types, related to the icosahedral configurations and amorphous structures, play an important role: (1) For Al, they represent 14.3% and 13.2% at 943K, respectively, and the two bond-types represent 27.5% of the total bond-types. It is worth noting that these percentages change with the system temperature. At 350K, the proportion of 1551 bond-type increases remarkably with decreasing temperature, reaching 29.4% of the total, whilst the 1541 bond-type only increase slightly to 14.6% of the total; the sum of the 1551 and 1541 bond-types makes up 44.0% of all bondtypes, indicating an increase of 16.5% from the corresponding proportion at 943K. (2) For Na, they represent 6.0% and 9.0% of all bond-types at 973K**,** respectively. While the sum of the 1551 and 1541 bond-types represents 45.8% of all bond-types, being increased about 30.8%. Highly interesting is that the relative numbers of 1551 bond-type is also increased remarkably with decreasing temperature, reaching 31.2% at 223K. From these results, it can be obviously seen that for the two systems, the 1551 bond-type plays a decisive role in the whole evolution process of microstructures.

For the relative numbers of the 1441, 1431, 1421 and 1422 bond-types related to the tetrahedral structures, the 1331, 1321, 1311 and 1301 bond-types related to the rhombohedral structures, and the 1661 bond-type related to hcp and bcc structures, are also similar to those obtained from previous works as above-mentioned.

Highly interesting is the 1771 bond-type, according to the definition of Honeycutt-Andersen bond-type indexes, it should possess seven-fold symmetry. It is well known that the seven-fold symmetry cannot exist in crystal solid state. However, in the Al system, although the relative number of 1771 bond-type is less than 0.1%, it still only exists in liquid and supercooled liquid states above 500K, and disappears in the solid state below 500K. This result just proves that the seven-fold symmetry cannot exist in crystal solid state, and further proves that the seven-fold symmetry also cannot exist in amorphous solid state. But for Na system, at 123K, it is still in the suppercooled liquid, so the 1771 bond-type can exist in it.


Table 1. Relations of the number of various bond types (%)of Al with temperature (K).

Firstly, the relative numbers of 1551 and 1541 bond-types, related to the icosahedral configurations and amorphous structures, play an important role: (1) For Al, they represent 14.3% and 13.2% at 943K, respectively, and the two bond-types represent 27.5% of the total bond-types. It is worth noting that these percentages change with the system temperature. At 350K, the proportion of 1551 bond-type increases remarkably with decreasing temperature, reaching 29.4% of the total, whilst the 1541 bond-type only increase slightly to 14.6% of the total; the sum of the 1551 and 1541 bond-types makes up 44.0% of all bondtypes, indicating an increase of 16.5% from the corresponding proportion at 943K. (2) For Na, they represent 6.0% and 9.0% of all bond-types at 973K**,** respectively. While the sum of the 1551 and 1541 bond-types represents 45.8% of all bond-types, being increased about 30.8%. Highly interesting is that the relative numbers of 1551 bond-type is also increased remarkably with decreasing temperature, reaching 31.2% at 223K. From these results, it can be obviously seen that for the two systems, the 1551 bond-type plays a decisive role in the

For the relative numbers of the 1441, 1431, 1421 and 1422 bond-types related to the tetrahedral structures, the 1331, 1321, 1311 and 1301 bond-types related to the rhombohedral structures, and the 1661 bond-type related to hcp and bcc structures, are also similar to those

Highly interesting is the 1771 bond-type, according to the definition of Honeycutt-Andersen bond-type indexes, it should possess seven-fold symmetry. It is well known that the seven-fold symmetry cannot exist in crystal solid state. However, in the Al system, although the relative number of 1771 bond-type is less than 0.1%, it still only exists in liquid and supercooled liquid states above 500K, and disappears in the solid state below 500K. This result just proves that the seven-fold symmetry cannot exist in crystal solid state, and further proves that the seven-fold symmetry also cannot exist in amorphous solid state. But for Na system, at 123K, it is still in the suppercooled liquid, so

Table 1. Relations of the number of various bond types (%)of Al with temperature (K).

From Table 1 and 2, it can be clearly seen that:

whole evolution process of microstructures.

the 1771 bond-type can exist in it.

obtained from previous works as above-mentioned.


Table 2. Relations of the number of various bond types (%) of Na with temperature (K).

On the whole, these simulation results are rather close to those obtained in our previous works on different-sized liquid metal systems (Liu R. S., et al., 1998, 1999, 2005a, 2005b; Dong K. J., et al., 2003) ; that is to say, for different-sized liquid metal systems, the simulation results of relative numbers of corresponding bond-types are similar to each other, there being only a minor difference during solidification processes.
