**A Molecular Dynamics Study on Au**

Yasemin Öztekin Çiftci1, Kemal Çolakoğlu1 and Soner Özgen2 *1Gazi University; Science Faculty, Physics Department, Ankara 2Firat University; Faculty of Art and Science, Physics Department, Elaziğ Turkey* 

### **1. Introduction**

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

Zhou, L. L., Liu, R. S., Tian, Z. A., et al.,(2011). Formation and evolution characteristics of bcc

metal Pb, *Trans. Nonferrous Met. Soc. China,* Vol. 21: 588-597.

phase during isothermal relaxation processes of supercooled liquid and amorphous

Theoretical and computational modeling is becoming increasingly important in the devolopment of advanced high performance materials for industrial applications.[1] Computer simulations on various metallic systems usually use simple pairwise potentials. However, the interactions in real metallic materials can not be represented by simple pairwise interactions only. A pure pairwise potential model gives the Cauchy relation, *C12*=*C44,* between the elastic constants, which is not the case in real metals. Therefore, manybody interactions should be taken into account in any studies of metals and metal alloys.

It is very important to calculate the phase diagrams of metallic systems and their alloys in order to achieve technological improvements. The phase diagrams are still obtained by using experimental techniques because there are no available methods for entirely theoretical predictions of all of the phase diagrams of any pure metal. Therefore, in the calculations of the phase diagrams some expressions have been formed by using theoretical or semi-empirical approach and their validity have been investigated in a selected portion of the phase diagrams. The expressions suggested in semi-empirical approaches generally contain some factors depending on temperature and pressure. Therefore, the calculated phase region is restricted by experimental limits. Today, the free energy concepts, such as Gibbs and Helmholtz, on the other hand, have been widely used to calculate the macroscopic phase diagrams [2, 3] in which thermodynamics parameters are dominant. In microscopic scale, their calculations require some vibrational properties which can be derived from elastic constants of the material. So, the correct calculations of the elastic constants are important as well as the calculations of phase diagrams.

MD simulations can be utilized to compute the thermodynamic parameters and the results of the external effects, such as temperature and pressure or stress acted on a physical system [4, 5]. In the MD simulations, the interatomic interactions are modeled with a suitable mathematical function, and its gradient gives the forces between atoms. Hence, Newton's equations of motion of the system are solved numerically and the system is forced to be in a state of minimum energy, an equilibrium point of its phase space. Although many properties of the system, such as enthalpy, cohesive energy and internal pressure, have been directly calculated in the MD simulations, the entropy which is required for the free energy calculations has not been directly obtained and it is possible to obtain it by some approaches involved harmonic and anhormonic assumptions. There are some investigations related to

A Molecular Dynamics Study on Au 203

between *i* and *j* atoms, *Fi*(ρ*i*) is the embedding energy to embed atom *i* in an electron density

In this work, we used a modified pairwise potential function in the framework of the Cai version [22] of the EAM. Recently, this potential function has been used by us for predicting several physical properties of some transitional metals [21,23-25]. The present form of the potential makes it more flexible owing to the constants, *m* and *n* in the multiplier forms. Such a factor included in the classical Morse function is treated by Verma and Rathore [26] to compute the phonon frequencies of Th, based on the central pair potential model. The

> ( ) ( ) *<sup>e</sup> r r <sup>e</sup> fr f e*

*n n*

*<sup>r</sup> <sup>m</sup> <sup>r</sup> <sup>r</sup> <sup>n</sup> <sup>r</sup>*

 

> 

*n e*

*ee e*

0 2 ( ) [1 ln ]

( 1) ( 1) <sup>1</sup> ( ) ( ) ( 1) ( ) *e*

*De r r e m r r r*

potential parameters and experimental input data for Au are given in Table 1.

*e*

where *α*, *β*, *D1* and *D2* are fitting parameters that are determined by the lattice parameter *a*0,

the host electron density at equilibrium state, *re* is the nearest neighbor equilibrium distance, and *F*0=*E*c*E*vf . In this potential model, there are four parameters: *β* and *D*1 are from twobody term, *m* and *n* are adjustable selected constants. The fitting parameters are determined by minimizing the value of exp exp <sup>2</sup> [( ) / ] *W XX X cal* . Here *X* represents the calculated and experimental values of the quantities taken into account in the fitting process. Hence, the potential functions can be fitted very well to the experimental properties of the matter, such as the vacancy formation energy (*E*v), cohesive energy (*E*c), elastic constants (*C*ij), and lattice constants (*a*0) in an equilibrium state. In the fitting process here, the cutoff distance is taken to be *r*cut=1.65*a*0. In the Eq. (3), the *f*e parameter is selected as unity for mono atomic systems because it is used for alloy modeling as an adjustable parameter to constitute suitable electron density. For the selected values of the constants *m* and *n*, the computed

The cohesive energy changes with the variation of lattice constants of Au calculated from Eq. (1) and from the general expression of the cohesive energy of metals proposed by Rose et al. [32] are compared in Fig.1. The Rose energy is also called as the generalized equation

<sup>0</sup> ( \*) (1 \*) *<sup>a</sup> E a E ae <sup>R</sup>*

*a E*

*a B*

0 \* 1/ <sup>9</sup>

*a*

\*

1/2

*C m*

(6)

(7)

*F F D* 

, (3)

, (4)

, (5)

, the elastic constants *Cij.* Here ρ*e* is

 

 

*e*

ρ*i*, and (*rij*) is the pairwise potential energy function between atoms *i* and *j*.

modified parts of the potential and the other terms are as follows:

the cohesive energy *E*c, the vacancy formation energy *E*v<sup>f</sup>

of state of metals and written as

these approaches: the calculation of the free energy between FCC and HCP structures [6, 7], the investigation of first order phase transition [8], the dependence of the phase diagram on the range of attractive intermolecular forces [9], the investigation of harmonic lattice dynamics and entropy calculations in metal and alloys [10], the calculation of the *P*-*T* diagram of hafnium [11], etc. Recently, the *P-T* diagrams for Ni and Al have been calculated by Gurler and Ozgen [12] by using the MD simulations based on the EAM technique [13].

The reliability of the results obtained from MD simulations depends on the suitable modeling of the interatomic interactions. Interatomic interactions are usually results of fits to various experimental data. However, it is not clear whether simulations performed at other temperatures still reproduce the experimental data accurately. Comparing theoretical and experimental elastic constants and other properties at various temperatures can serve as a measure of reliability and usefulness of potential models [14, 15]. In fact, there are several potential energy functions that can be used for the metallic systems. However, the EAM, originally developed by Daw and Baskes [16, 17] to model the interatomic interactions of face-centered cubic (FCC) metals, has been successfully used to compute the properties of metallic systems such as bulk, surface and interface problems. The reliability of the EAM in the bulk and its simple form for use in computer simulations make it attractive.

When a liquid metal is quenched through the super-cooled region, a phase transition from liquid to glass takes place. Several techniques have been proposed to obtain a disordered state [18-20]. Among them the rapid solidification method is widely used for the amorphous phase. However, due to the demand of a high cooling rate this method is restiricted in most experimental cases. Thus, the computer simulation of molecular dynamics is applied.

In this study, in order to model Au metallic systems we have used the EAM functions modified by us (Ciftci and Colakoğlu [21]), developed firstly by Cai [22]. In this work, we have carried out MD simulations to obtain the *P*-*V* diagrams at 300 K and the *P*-*T* diagrams of the systems for an ideal FCC lattice with 1372 atoms, by using an anisotropic MD scheme. In addition, the bulk modulus and specific heat of the system in solid phase are determined and results-driven simulations are interpreted by comparing with the values in literature. We have also calculated the pressure derivatives of elastic constants and bulk moduli for Au. The obtained results are compared with the values in the literature. The another purpose of this work is to explore the glass transition and crystallization of Au using EAM .
