A Molecular Dynamics Study

G.U. Journal of Science 22(1): 15-19 (2009) www.gujs.org Structures and Energetics of Cu21-Cu71 Clusters: A Molecular Dynamics Study Saime Şebnem ÇETĐN1, Süleyman ÖZÇELĐK1♠, Ziya B. GÜVENÇ2 1 Gazi University, Faculty of Arts & Sciences, Department of Physics, 06500, Ankara, TURKEY 2 Çankaya University, Faculty of Engineering & Architecture, Department of Electronic and Communication Engineering, 06530, Ankara, TURKEY Received: 24.06.2008 Revised: 22.12.2008 Accepted: 31.12.2008 ABSTRACT Using Molecular Dynamics and thermal quenching simulations the stable geometrical structures and energies of Cun (n=21-71) clusters are identified. The interaction between the cluster atoms is modeled by an EmbeddedAtom Potential Surface, Voter and Chen’s version. The stable geometrical structures and energies are obtained from 500 phase space coordinates generated along high-energy trajectories. The internal energy (about T=2500 K) is above the melting temperature of the Cun clusters. The thermal quenching technique is employed to remove slowly the internal kinetic energy of the clusters. Because of this slow minimization process the locally stable isomers are separated from those meta-stable ones. Key Words: Cu clusters, Cluster Structures, Molecular Dynamics, Computer Simulations. 1. INTRODUCTION Geometrical structures and energetics of the clusters are important factors for determining their chemical and physical properties. Therefore, the structure and dynamics of small clusters, in particular transition metal clusters, have been attracted much attention theoretically [1-16] and experimentally [17-20] in the last decade. Theoretical calculations can complement such experimental investigations using ab initio methods and/or sufficiently accurate interaction potentials in dynamical simulations to investigate the structure, dynamics, and reactivity of clusters as a function of cluster size. A new scientific filed so-called “nano-science” has been formed by these activities. Small copper clusters with up to 5 atoms [1] and 10 atoms [2] were studied by using density functional theory. Decay pathways and dissociation energies of Cu+ (2≤ n ≤25) cluster were studied experimentally [19]. Structures and stability of up to 56 atoms of the copper were employed by Darby et al.[21] using the many body Gupta potential. In Ref. [14], original version of the Embedded Atom Model (EAM) developed by Foiles et al. [16] for fcc metals was employed in their Molecular Dynamics (MD) study. In Ref. [14] however, Voter and Chen's version of the ♠ Corresponding author, e-mail: EAM potential was used to study structures and binding energies of the lowest energy isomers, and melting behavior of the Cun, (n=2-23) clusters. We have also used Voter and Chen’s version of the EAM (see for details Ref.[22]) in our work. Our focus is on obtaining the number of stable isomers, average bond lengths, and magic sizes of the Cun, (n=21-71) in addition to the lowest energy structures and energetics of the clusters. In Section 2 the detail of computational procedure is given. The results and their analyses are discussed in Section 3, and we conclude with a brief summary. 2. THE POTENTIAL AND COMPUTATIONAL PROCEDURE Because of the fitting procedure mentioned above, we have incorporated the EAM [4] in our studies since such fitting may increase the validity of this potential in the finite size range. The stable geometrical structures of the Cun, n=21-71, clusters are identified using MD and thermal quenching (TQ) simulations. The clusters are prepared initially with zero total linear and angular moment. After that their internal energies are increased to about T = 2500K. These energies are much higher than the “melting” temperatures of the Cun clusters. Along the high-energy trajectories in phase space 500 16 G.U. J. Sci., 22(1):15-19 (2009)/ Saime Şebnem ÇETĐN, Süleyman ÖZÇELĐK♠, Ziya B. GÜVENÇ independent set of phase space coordinates are selected. Hamilton’s equations of motion were solved for all the atoms in the cluster using Hamming’s modified 4th order predictor-corrector algorithm with a step-size of 1x10-15 s. For each set of the phase space coordinates the TQ technique is used to remove the internal kinetic energies of the clusters (the internal kinetic energy of a cluster is set to zero at every 50 simulation steps, and the process continues until the energy is completely removed). Because of this slow minimization process, clusters cannot stay at the meta-stable locations of the potential energy surface (PES), and finally, they will be trapped at the bottom of the PES “wells”. Thus, the locally stable isomers are separated from meta-stable ones. 3. RESULTS AND DISCUSSION The geometries of the most stable isomers of Cun (n = 21-71) clusters, average bond lengths of their stable isomers and energetics, average interaction energy per atom of these clusters are obtained. The minimum energy geometries and energetics of the most stable copper clusters are given in Figures 1 and 2. As seen the Cu55 has a structure of shell form (Ih) [21]. The Cu56 is simply formed by capping one of the faces of this icosahedral structure. Cu21 Cu22 Cu23 Cu24 Cu25 Cu26 Cu27 Cu28 Cu29 Cu30 Cu31 Cu32 Cu33 Cu34 Cu35 Cu36 Cu37 Cu38(1) Cu38(2) Cu39 Cu40 Cu41 Cu42 Cu44 Cu45 Cu46 Cu43 Cu47 Figure 1. The Minimum energy geometries of the most stable copper clusters n=21-47. G.U. J. Sci., 22(1):15-19 (2009)/ Saime Şebnem ÇETĐN, Süleyman ÖZÇELĐK♠, Ziya B. GÜVENÇ 17 Cu51 Cu48 Cu49 Cu50 Cu52 Cu53 Cu54 Cu55 Cu58 Cu59 Cu57 Cu56 Cu60 Cu61 Cu62 Cu63 Cu64 Cu65 Cu66 Cu67 Cu68 Cu69 Cu70 Cu71 Figure 2. The minimum energy geometries of the most stable copper clusters n=48-71. G.U. J. Sci., 22(1):15-19 (2009)/ Saime Şebnem ÇETĐN, Süleyman ÖZÇELĐK♠, Ziya B. GÜVENÇ The average nearest-neighbor distance for the Cu clusters n = 21-71 vary between 2.5143Å and 2.5339Å. As the cluster size increases, the average nearestneighbor distance approaches the bulk value (the bulk value is 2.56Å for the fcc copper crystal [23]). Obtained results for binding energies from10 up to 56 atoms of the cluster are agree with calculated values by Darby et al [21] using many body Gupta potentials for these size of the copper clusters. In order to investigate the relative stabilities of the clusters we consider here the evolution of the binding energy per atom (the average interaction energy), Ea, the first difference energy, ∆E(1), and the second difference energy, ∆E(2), which are defined in terms of the total interaction energy of the cluster (Figures 3-5). 3.0 Second Difference Energy(eV/Atom) 18 n=53 n=55 2.0 (c) 1.0 n=29 n=45 n=38 n=59 n=66 0.0 -1.0 -2.0 -3.0 21 31 41 51 61 71 Size,n Figure 5. The second difference energy ∆E(2)=En+12En+En-1 as a function of the cluster size, n. Binding Energy (eV/Atom) -2.6 These energies are defined in terms of the total interaction energy of the cluster (...truncated)