Molecular-dynamics simulations of silver clusters

Physica E 5 (1999) 1–6

www.elsevier.nl/locate/physe

Molecular-dynamics simulations of silver clusters Sakir  Erkoc ∗ , Tugrul Ylmaz Department of Physics, Middle East Technical University, 06531 Ankara, Turkey Accepted 18 August 1999

Abstract Structural stability and energetics of silver clusters, Agn (n = 3–177), have been investigated by molecular-dynamics simulations. An empirical model potential energy function has been used in the simulations. Stable structures of the microclusters with sizes n = 3–13 and clusters generated from FCC crystal structure with sizes n = 13–177 have been determined by molecular-dynamics simulation. Five-fold symmetry appears on the spherical clusters. ? 1999 Elsevier Science B.V. All rights reserved. PACS: 36.40.−c; 36.40.Qv; 61.46.+w; 02.70.Ns Keywords: Silver clusters; Empirical potentials; Molecular dynamics

1. Introduction Clusters play an important role in understanding the transition from the microscopic structure to the macroscopic structure of matter. The research eld of clusters, particularly microclusters, has shown a rapid development in both experimental and theoretical investigations in the last two decades [1–3]. Although there are considerable improvement in the experimental techniques, there are still diculties in the production and=or investigation of isolated microclusters of some elements. Computer simulations provide help for a deeper understanding of the experimental observations on the one hand, and on the other, they can ∗ Corresponding author. Tel.: +90-312-210-32-85; fax: +90-312-210-12-81. E-mail address: [email protected] (S.  Erkoc)

also be applied for the systems which are practically dicult to conduct experiments on. Atomistic level computer simulations using empirical model potentials have been used successfully to investigate bulk, surface, and cluster properties of elements. Several empirical potential energy functions have been proposed and applied to various systems in the last decade [4]. Silver microclusters are interesting and have potential importance in the physics and chemistry of transition metals and transition metal alloys (see, for example, Ref. [5]). Silver clusters are of special interest because of their practical applications in catalysis and photography [6,7]. There exist many experimental [8–12] and theoretical [13–34] studies of the coinage metal (Cu, Ag, and Au) clusters. In particular for silver microclusters there are quantum mechanical calculations published for dimers and trimers [14 –17,19,21,22], for tetramers

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and pentamers [16,17,21–23], for hexamers [24,25], and for microclusters up to 9 atoms [26] by several research groups. There are also several empirical potential energy function calculations [27–33] for silver microclusters up to 7 atoms, and an empirical potential parametric calculation [34] for some model silver clusters up to 309 atoms. In this work we have investigated the structural properties of isolated silver clusters containing 3–177 atoms. Making use of a recently developed empirical potential energy function (PEF) for silver [32] we performed molecular-dynamics (MD) simulations to predict the stable structures of silver clusters. 2. The potential energy function Total interaction energy () of an N particle system may be calculated from the sum of eective-pair interactions [32]: =

N P i¡j

Ue (rij );

(2.1)

where the eective-pair potential energy function, Ue (r), is in the form [32] 2 A1 −1 r 2 A2 e + D22 2 e−2 r : (2.2)  1 r r The potential parameters used for silver are: A1 = 220:262366, A2 = − 26:0811795, 1 =1:72376253, 2 = 1:81484791; 1 = 0:673011507; 2 = 0:120620395, D21 = 1:00610152, and D22 = 0:221234242 [32]. In these parameters, the energy is in eV and the distance  The eective-pair potential, Ue (r), with D21 = is in A. D22 = 1:0 represents the dimer potential, U (r). The present empirical PEF for the silver element satis es the dimer potential U (r), the bulk cohesive energy, [Eq. (2.1)], and the bulk stability condition 9=9V = 0, exactly [32]. The present PEF also satis es the crystal stability; it gives the FCC crystal structure as the most stable. Calculated elastic constants are in reasonable agreement with the experimental values [32]. Using this PEF we have performed MD simulations to obtain the stable structures of silver microclusters with the number of atoms from 3 to 177. Similar PEF with dierent parameter sets was successful to simulate bulk and microcluster properties of Cu, Ag and Au elements with the number of atoms from 3 to 7

Ue (r) = D21

[32], and nanowire properties of Cu [35]. We expect that we may also use this PEF for small clusters of silver. In this work we extend the previous silver cluster calculations, which were done for the number of atoms 3–7 by static minimization [32], by performing MD simulations. 3. Results and discussions The silver clusters containing 3–177 atoms have been investigated by performing MD simulation at constant temperature to obtain the stable structure of each cluster. In the MD simulations the empirical PEF, Eq. (2.1), is used. Nordsieck–Gear algorithm [36,37] with predictor–corrector method [38] is used in the solutions of the equations of motion. Simulations are carried out by starting at 1000 K, and then temperature is reduced upto 1 K in four stages namely, 1000, 100, 10, and 1 K. We have done this procedure to be able to increase the probability of obtaining the stable con guration of the potential energy surface of the cluster simulated. The time step has been taken as 1:0 × 10−15 s. In the simulations we have taken the number of MD steps at most 50 000 for the clusters with the number of atoms from 3 to 13. In the last 10 000 steps the temperature rescaling was turned o and the system relaxed to reach the thermal equilibrium. For the clusters with the number of atoms n ¿ 13, the number of MD steps was taken as 80 000, again in the last 10 000 steps the system was allowed to relax and reach the thermal equilibrium. That many steps were enough to reach equilibrium in total energy and to attain the thermal equilibrium of the system studied. We performed this procedure for every temperature reduction in the simulations. Like conventional molecules, most clusters, in general, have some well-de ned geometry corresponding to the absolute minimum energy of their potential surfaces. There might be many local minima on the potential energy surface of a many-particle system and also there might be many isomers of a many-particle system. In the present study, we have generated the microclusters starting from 3 particles. After getting the stable structure by temperature reduction procedure we added one atom to obtain the next cluster, and repeated the same procedure. We applied this method for the microclusters with sizes

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Fig. 1. (a) Equilibrium structures of silver microclusters with sizes n = 3–13. Ideal (unrelaxed) (i) and equilibrated (relaxed) (r) spherical clusters generated from FCC crystal structure with sizes (b) n = 13–79, and (c) n = 87–177.

n = 3–13. The stable structures of silver microclusters with sizes n = 3–13 obtained by the MD simulations are shown in Fig. 1a. These structures represent the con guration of the system studied at the last MD step. We have also simulated spherical silver clusters that are generated from the FCC crystalline structure by taking the rst, second and so on up to ninth neigh-

bors to a central atom in the bulk. The cluster models generated in this form contain the number of atoms n = 13; 19; 43; 55; 79; 87; 135; 141; 177. We applied the same simulation procedure for these spherical clusters. The three-dimensional structures of these clusters are presented in Figs. 1b and c. In Figs. 1b and c we have displayed the ideal and the relaxed

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spherical silver clusters to see the dierence in the structures. The stable structure for the cluster n = 13 obtained from the two models gives the same geometry, that is, the stable structure obtained from the relaxation of the spherical cluster obtained from FCC is the same as the one obtained from the one-by-one generation. The binding energy, i.e., the average interaction energy per atom in the cluster versus the cluster size, the number of atoms in the cluster, is plotted for the stable structures in Fig. 2. The average binding energy per atom decreases as the cluster size increases. It shows an exponential like decaying. From the energetics point of view we may classify the silver clusters studied into three groups (3–13; 13–55; 55 –177). In the rst group of clusters the variation of average binding energy with respect to the cluster size is fast. However, the variation of the average binding energy with respect to the cluster size for the second group of clusters is relatively slow. This trend appears in the third group of clusters even more slowly. The general decaying behavior in the average binding energy with respect to the cluster size is common almost for all metal clusters [3]. The size dependence of the average binding energy per atom for metal clusters is predicted by the empirical relation [2] Eb (N ) =  + 21=3 ( 12 De − )N −1=3 ;

(3.1)

here  is the bulk cohesive energy of the element forming the cluster, and De is the dimer binding energy. For silver  = −2:95 eV=atom [39] and De = −1:65 eV [40]. Fig. 2 contains also the binding energy variation with respect to the cluster size using Eq. (3.1). The functional dependence of both predicted from Eq. (3.1) and present calculation looks similar, but there is a shift in the energy. Since the present PEF was scaled to the bulk cohesive energy in its parametrization, as n → ∞ this curve goes asymptotically to the bulk cohesive energy value of −2:95 eV. However, it is not possible to reach the bulk value for nite systems due to surface eects; there is always surface in nite systems and missing interactions decrease (in magnitude) the total energy. On the other hand, for isolated clusters the average binding energy per atom in the cluster, Eb = =N , may also be expressed as a function of cluster size, N , [3], Eb = Ev + Es N −1=3 + Ec N −2=3 ;

(3.2)

where the coecients Ev ; Es , and Ec correspond to the volume, surface, and curvature energies of the particles forming the cluster, respectively. The linear t of the data to Eq. (3.2) gives Ev = −1:377 and Es = 1:774, the quadratic t gives Ev = −1:465; Es = 2:284, and Ec = −0:626. The volume energy term should be equal to the bulk cohesive energy value of −2:95. The reason for the dierence between the calculated value from the t and the experimental value is that the clusters considered in the present study are not large enough. As the cluster size increases the calculated volume energy approaches the bulk cohesive energy. The stable structures of the microclusters with sizes n = 3–7 have a regular symmetry. The corresponding point groups of these clusters are: Ag3 (D3h ); Ag4 (Td ); Ag5 (D3h ); Ag6 (Oh ); Ag7 (D5h ). On the other hand, clusters with sizes n = 9–12 seem to be formed by adding atoms one by one to the seven-atom cluster. In these structures the con guration of seven-atom cluster keeps its symmetry. However, eight-atom cluster does not show this property, there is no pentagonal bipyramidal seed in the eight-atom cluster. The Ag13 has the symmetry Ih . All the quantum calculations give the stable structure of trimer as trigonal, and the structure of tetramer as rhombus. All-electron spin density approach [25] gives silver clusters upto six atoms as planar (two-dimensional). However, complete active space self-consistent eld (CASSCF) procedure [24,26] gives the stable structure as pentagonal pyramid for six-atom cluster. CASSCF calculation [26] gives seven-atom cluster as pentagonal bipyramid, and eight- and nine-atom clusters same as found in this work. Many-body embedded-atom potential calculation [33] gives the stable structures of silver clusters upto eight atoms same as found in this work. Previous empirical potential calculations [27–32] give the similar structures found in this work. In another empirical potential calculation [34] the icosahedron structures of the spherical silver clusters with sizes n = 13; 55; 147; 309 are found as more stable with respect to decahedron and cuboctahedron structures. In Ref. [34], calculations were carried out parametrically. Here some of the potential parameters were changed to nd minimum energy con guration with respect to the starting geometry. On the other hand, in Ref. [34] the minimization procedure was performed

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Fig. 2. Average interaction energy per atom versus cluster size, N . The squares represent the ideal spherical cluster energies, the circles represent the relaxed cluster energies, and the triangles represent the energies obtained from Eq. (3.1).

by a function minimization method, conjugate gradient method, a sort of static minimization. In the present work we have performed MD procedure without the potential parameters. The minimization procedure used in the present work may be considered as a dynamic minimization. As we have mentioned in the rst paragraph of this section, we relaxed the system at a predetermined temperature, this allows the system to go to the possible minimum energy con guration, because particles move under the eect of the force due to the interparticle interactions. The equations of motion of the particles were solved using the Nordsieck–Gear algorithm [36,37] within the seventh-order predictor–corrector method [38]. Undoubtedly, dynamic minimization simulates the system more realistically. The bond lengths (or the average nearest-neighbor  at the last MD step for distances) vary around 3:0 A cluster sizes n = 3–19. As the cluster size increases the average nearest-neighbor distance aproaches the bulk value of dnn (dnn is the nearest-neighbor distance in the  for silver crystal in FCC struccrystal, dnn = 2:89 A ture [39]) for the atoms within the volume of the cluster. On the other hand, the nearest-neighbor distance

between the atoms on the surface region vary around  In general, the nearest-neighbor distances are 3:0 A. relatively larger than the dimer distance re (the dimer  for silver-dimer [40]). The spherdistance re = 2:48 A ical clusters keep their spherical form after the relaxation, but atoms on the surface region reconstruct slightly with respect to the original positions. Five-fold symmetry appears on the surface region of the relaxed spherical clusters. This situation is clearly seen in Figs. 1b and c. Since the eective pair potential parameters were determined using crystal structure information, the estimated average interatomic distances in the clusters might be slightly larger than the actual values. Summarizing, we have investigated the structural stability and energetics of isolated silver clusters containing 3–177 atoms. We nd that silver microclusters prefer to form three-dimensional compact structures. On the other hand, ve-fold symmetry appears on the surface region of the spherical clusters. Acknowledgements One of the authors (SE)  would like to thank METU for partial support through the project

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METU-AFP-98-01-05-03 and TUBITAK (Scienti c and Technical Research Council of Turkey) through the project TUBITAK-AY=194. References [1] S. Sugano, Y. Nishina, S. Ohnishi (Eds.), Microclusters, Springer, Berlin, 1987. [2] H. Haberland (Ed.), Clusters of Atoms and Molecules, Springer, Berlin, 1994. [3] G. Scoles (Ed.), The Chemical Physics of Atomic and Molecular Clusters, North-Holland, Amsterdam, 1990. [4] S.  Erkoc, Phys. Rep. 278 (1997) 79. [5] V. Bonacic-Koutecky, P. Fantucci, J. Koutecky, Chem. Rev. 91 (1991) 1035. [6] J. Flad, H. Stoll, H. Preuss, Z. Phys. D 6 (1987) 193. [7] J. Flad, H. Stoll, A. Nicklass, H. Preuss, Z. Phys. D 15 (1990) 79. [8] C. Jaackschath, I. Rabin, W. Schuze, Z. Phys. D 22 (1992) 517. [9] J. Ho, K.M. Ervin, W.C. Lineberger, J. Chem. Phys. 93 (1990) 6987. [10] G. Gantefor, M. Gausa, K.H. Meiwes-Broer, H.O. Lutz, J. Chem. Soc. Faraday Trans. 86 (1990) 2483. [11] O. Chesnovsky, K.J. Taylor, J. Conceicao, R.E. Smalley, Phys. Rev. Lett. 64 (1990) 1785. [12] G. Alameddin, J. Hunter, D. Cameron, M. Kappes, Chem. Phys. Lett. 192 (1992) 122. [13] J. Flad, G. Igel-Mann, H. Preuss, H. Stoll, Chem. Phys. 90 (1984) 257. [14] W. Andreoni, J.L. Martins, Surf. Sci. 156 (1985) 635. [15] C.W. Bausclicher Jr., S.R. Langho, P.R. Taylor, J. Chem. Phys. 88 (1988) 1041. [16] C.W. Bausclicher Jr., S.R. Langho, H. Partridge, J. Chem. Phys. 91 (1989) 2412. [17] C.W. Bausclicher Jr., S.R. Langho, H. Partridge, J. Chem. Phys. 93 (1990) 8133.

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