ARTICLE IN PRESS

Physica B 404 (2009) 205–209

Contents lists available at ScienceDirect

Physica B

0921-45

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/physb

Molecular-dynamics investigation of structural transformations of a Cu201

cluster in its melting process

Lin Zhang �, Cai-Bei Zhang, Yang Qi

College of Science, Northeastern University, P.O. Box 104, Shenyang 110004, China

a r t i c l e i n f o

Article history:

Received 4 June 2008

Received in revised form

1 October 2008

Accepted 15 October 2008

PACS:

31.15.Qg

36.40.�c

61.46.Bc

Keywords:

Cluster

Molecular dynamics

Computer simulation

Surface

26/$ - see front matter & 2008 Elsevier B.V. A

016/j.physb.2008.10.028

esponding author. Tel.: +86 24 83678479; fax

ail address: zhanglin@imp.neu.edu.cn (L. Zhan

a b s t r a c t

We perform molecular-dynamics calculations to investigate the structural transformation of a copper

cluster containing 201 atoms in its melting process within the framework of the embedded-atom

method (EAM). Concerning melting, the obtained results reveal that its structural changes are different

from those of larger-size clusters containing several hundreds or more atoms and smaller-size clusters

containing tens of atoms. The melting process of this Cu201 cluster involves three stages, firstly some

atoms in inner regions of this cluster move into outer regions accompanying the structural

transformation of the local atom packing, followed by the continuous interchange of atomic positions,

and finally this cluster is wholly disordered. During the temperature increase, the structural changes of

different regions determined by atom density profiles result in apparent increases in internal energy. By

decomposing peaks of pair distribution functions (PDFs) according to the pair analysis (PA) technique,

the local structural patterns are identified for the melting of this cluster.

& 2008 Elsevier B.V. All rights reserved.

1. Introduction

As a bridge between isolated atoms and bulk materials,metallic clusters have attracted much attention of researchers inthe fields of physics, chemistry, and materials due to theirimportance in catalysis and surface nanotechnology [1–8].Previous studies have shown that the most important chemicaland physical properties of these clusters are greatly dependent ontheir geometrical structures and sizes. Those studies also demon-strated that the transition from solid to liquid for these clusterstakes place over a finite temperature range, and meltingtemperatures of the metal particles decreased with a reductionof their size [9–11]. The main cause for these is the high surface tovolume ratio for these particles, which as a consequence of theimproved free energy at the particle surface results in a decreaseof the melting point. For the clusters containing several hundredsto thousands of atoms, the melting process can be described asinitially premelting of the surface and then the overall melting ofthe whole clusters [12]. Melting behaviors of small-size clusterscontaining tens of atoms present different patterns. For example,

ll rights reserved.

: +86 24 83683674.

g).

for Cu54–Cu56 clusters having icosahedron-based geometries, thesimulation results show that the melting processes of the Cu54

and Cu55 clusters begin from movements of surface atoms intoinner parts of these clusters, whereas the Cu56 cluster has initiallocal disorder in sub-surface atoms [13]. Nevertheless, thestructural change mechanism of the clusters containing about100–200 atoms on heating is still poorly understood. For studyingthe microscopic details of these systems, molecular dynamics(MD) is a well-established technique because of lack of appro-priate experimental techniques and prohibitive computationalexpense from sophisticated ab initio approaches. In MD simula-tions, empirical potentials based on the embedded-atom method(EAM) are adequate to characterize these face-centered cubic(FCC) metallic systems, and some EAM versions have beenemployed to describe the properties of free clusters and clusterssupported on surfaces of the same kind of material for FCCtransition metals, such as Cu systems [14–21].

The purpose of this paper is to investigate the structuraltransformation of a copper cluster containing 201 atoms in itsmelting process. The whole melting process is traced from theatomic energy and local structural changes. How the structuralchanges accompanying the atom packing affect the internalenergy is demonstrated by decomposing the pair distributionfunctions (PDFs) according to the local environment of the pairs ofEAM atoms. Here, regions with multilayer structures of the

ARTICLE IN PRESS

L. Zhang et al. / Physica B 404 (2009) 205–209206

studied cluster are determined by atom density profile calcula-tions.

-2.95

-2.90

2. Model and simulation

All the atoms in the simulation are assumed to interact via anEAM potential of Cu constructed by Mei et al. [22], and atomictrajectories are generated in the canonical ensemble MD atelevated temperatures. A time step of 1.6�10�15 s is used in thecalculations. The lattice constant a0 of bulk Cu in the simulationsis 3.615 A.

Initially, we construct a 20a0�20a0�20a0 bulk FCC Cu crystal,then a crystal fragment, containing 201 atoms, of a sphericalshape covered by (10 0) and (111)-like facets is extracted fromthis constructed FCC crystal and kept in a MD simulation cell,where the volume of the MD simulation cell V is equal to20a0�20a0�20a0. Periodic boundary conditions are used in thesimulations. As shown in Fig. 1, O is the center of this Cu201 cluster,and r the radial direction. The simulations are performed bystarting with the optimal structure at 300 K, then increasing thetemperature gradually to 800 K at an increment of 50 K corre-sponding to the process at a heating rate of 6.25�1010 K/s, wherethe heating rate is calculated as the temperature interval dividedby the simulation time. At each temperature, the initial runs takeabout 480,000 time steps to reach equilibration, and subsequent20,000 time steps to record the atomic trajectories that are usedto study the structural properties of the simulated system. Thetemperature is kept constant by rescaling the atomic velocitiesevery time step.

In calculating the structural changes of this cluster, thefollowing values are determined [23]:

gLgðrÞ ¼V

N

Xi2Lg

Xjai2Lg

dð r*� r*

ijÞ

* +, (1)

rðriÞ ¼ hNii=N, (2)

where /S denotes the average over the entire trajectory, gLg(r) isthe PDF in the layer Lg, V the volume of the simulated MD cell, andN the number of atoms in this cell. The density profile r(ri) iscalculated by dividing the system into some layers and accumu-lating a histogram of the number, where /NiS is the averagenumber of atoms in layer i; and ri is taken as the center of thislayer.

In this study, four integers are used to perform local structureanalyses or pair analysis (PA) that Honeycutt and Andersen [24]specified to study structural changes of small Lennard-Jonesclusters during melting and freezing [25]. The first integerindicates whether or not the pairs of atoms referred to are nearneighbors or, equivalently, are considered to form a bond in agiven cutoff distance. If they are bonded, the integer is 1. Thesecond integer is the number of neighbors common to the bondedpair atoms, and the third is the neighbor relationships among the

Fig. 1. Geometry of the Cu201 cluster at 300 K.

shared neighbors. The fourth integer is arbitrarily added toprovide a unique arrangement of the pairs. For example, 1551pairs correspond to two bonded atoms with five commonneighbors that form a pentagon of near-neighbor contacts, andthis type of pairs is characteristic of icosahedral ordering; 1421and 1422 pairs can be found greatly in FCC-like and HCP-like localstructures.

3. Results and discussion

To study the stability of the cluster structure against thermalagitation, Fig. 2 plots the temperature variation of the systeminternal energy per atom from 300 to 800 K. As plotted in thisfigure, this energy curve exhibits some characteristics that thereare two separate linearly increasing ranges at 300–550 K andabove 700 K, where the temperature range above 700 K has ahigher slope than its counterpart at 300–550 K. One fluctuatingincrease range occurs at 550–700 K, implying a drastic structuretransformation corresponding to the melting process. In thisfluctuating range, three separate increase stages correspond totemperature ranges of 550–600, 600–650, and 650–700 K, andone abrupt increase of internal energy can be observed between650 and 700 K. The following atomic density profiles andcorresponding atomic packing will be used to illustrate thestructural changes at these temperature ranges.

In Fig. 3, left parts are atom density profiles of this Cu201

cluster at 300, 550, 600, 650, 700, and 800 K along the r directionas shown in Fig. 1. The profile at 300 K is composed of 11 shellscorresponding to 11 intensity peaks labeled from 1 to 11 aspresented in this figure. The widths of these peaks result fromthermal movements of atoms around their equilibrium positions.Here, we separate this cluster into three regions, including peaks1–4, labeled region I, peaks 5 and 6, region II, and peaks 7–11,region III. With increasing temperature, although the atoms stillmove around their equilibrium positions, peaks 11 and 7 in theregion III are almost combined into peaks 10 and 8, and peak 9disappears. In addition, the peak heights of the density profile at550 K are lower and the widths are broadened. As the temperatureis increased to 600 K, the profile is different from those at 300 and550 K, and it gives us a suggestion that structural changes haveoccurred and as a whole this cluster has been deformed. Onincreasing the temperature further, the features of density profilesexhibit some low and broad peaks observed in the three regions.When the temperature is increased up to 800 K, three separate

300-3.15

-3.10

-3.05

-3.00

Eav

/ev

T /K400 500 600 700 800

Fig. 2. Variation of internal energy per atom in the cluster with temperature.

ARTICLE IN PRESS

Fig. 3. Atomic density profiles and the corresponding atomic configurations of the cluster at 300, 550, 600, 650, 700, and 800 K.

L. Zhang et al. / Physica B 404 (2009) 205–209 207

broad peaks in the three regions can be observed, which is a proofthat the atom packing becomes disordered at this temperature.

Right parts of this figure show the atom packing in the threecolumns labeled as I–III corresponding to the three regions I–III,respectively, determined by the atom density profiles, where theatom packing is recorded from the average positions of the atomsin the last 20,000 time steps. As shown in these figures all atomsin this cluster preserve their FCC structures up to nearly 600 K. At

300–550 K, there are 43, 92, and 66 atoms, respectively, in theinner, near-surface, and surface regions, respectively, labeled I–III.As the temperature is increased, the deformation of this clusterinvolves mostly atoms accompanying interchange of positionsfrom the inner region to outer regions, including near-surface andsurface regions, and local structural changes, whereas some atomsin the surface region are locally ordered. The thermal expansionfrom the atom movements contributes to a distribution of atoms

ARTICLE IN PRESS

300

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Rel

ativ

e nu

mbe

r of p

airs

%

T /K

1311142114221551

400 500 600 700 800

Fig. 5. Relative number of atom pairs as a function of temperature.

L. Zhang et al. / Physica B 404 (2009) 205–209208

in a larger space as shown in the density profile at 600 K. Hence,the structural changes result in an apparent energy increasebetween 550 and 600 K as plotted in Fig. 2. At 600 K, the numbersof atoms in these three regions are changed to 34, 82, and 85,respectively, because of the atom movements. On increasing thetemperature to 650 K, although the number of atom in the firstregion is still 34, continuous interchange of positions amongatoms results in the movements of seven atoms in the secondregion into the third region. Therefore, in this temperature range,the energy plot presents a slow increase. However, in thetemperatures range 650–700 K, a dramatically increasing stageof energy exists as plotted in Fig. 2. In this stage, atoms in theouter regions move into the inner region, and the number ofatoms in the first region is increased to 38. Meanwhile, thenumbers of atoms in the second and the third regions are changedto 80 and 83, respectively. In the high-temperature range700–800 K, more atoms in the surface region move into the innerregions, including the first and second regions, contributing to anearly linear behavior of energy. At 800 K, the numbers of atomsin the first and second regions are separately increased to 39 and102, respectively. Here, the atom density profiles and thecorresponding atomic configurations present structural differ-ences at different temperatures, but they cannot describe thedetailed local structural evolution of this cluster. The followingPDFs and corresponding PA will be used to illustrate thisevolution.

PDFs of the simulated cluster at some temperatures between300 and 800 K are presented in Fig. 4. The PDF at 300 K has atypical crystal characteristic, where there is one main peak as wellas some other small peaks, and the main peak of this PDF is farhigher than the other peaks, which implies that most of atompairs are populated in the first peak. As the temperature isincreased to 550 K, the peaks of the PDF decrease in height,resulting from increasing thermal movements. At 600 and 650 K,the PDFs exhibit different behaviors from those at 300 and 550 K.Besides an apparent decrease of the first peaks, the second peaksdisappear, the heights of the third peaks decrease, which combinewith the fourth peak, the sixth peaks broaden, and the other peaksdisappear. These differences imply that structural changes occurin the temperature range 550–650 K. When the cluster is heatedabove 700 K, the PDFs show disordered behaviors, where thesecond and the fourth peaks disappear and the third peaks are

0.50

30

60

90

120

150

800K

700K

650K

600K

550K

300K

g (r

)

r /a0

1.0 1.5 2.0 2.5 3.0

Fig. 4. PDFs of the Cu201 clusters at 300, 550, 600, 650, 700, and 800 K.

also broadened. The following PA analysis will be used to describethe differences in local structures at these temperatures.

Fig. 5 shows the relative numbers of bonded atom pairs,including 1311, 1421, 1422, and 1511 pairs, as a function oftemperature during the heating of the cluster, corresponding tothe first peaks of PDFs as presented in Fig. 4. In this figure, therelative number is the ratio of the number of one kind of atomicpairs to the sum of atomic pairs, and it can be used to determinethe ordered degree of this cluster at elevated temperatures. Below550 K, accounting for the high ratios of surface atoms in thiscluster contributing to the 1311 pairs, the 1421 pairs take up about65% of the total pairs, which proves that the cluster has the FCCstructure. At 550–600 K, there exist decreases of the 1421 and1311 pairs accompanying two newly formed types of the 1422pairs and 1551 pairs in this cluster. These new pairs give us animplication that the atomic interchange movements haveoccurred drastically and changed the local structures of theseclusters. Especially regarding the 1421 pairs, their abrupt decreasesuggests that the whole cluster has been deformed and the FCCstructure is being destroyed. As the temperature is increasedabove 600 K, all the pairs begin to change, and either increase ordecrease. At 600–650 K, it can be observed that when the 1421pairs keep their decreasing trend, both the 1422 and 1551 pairsare also decreased, except for the increase of the 1311 pairs. In thefollowing temperatures, the 1311, 1421, and 1422 pairs presentdecreasing behaviors, and when the temperatures are higher than700 K, the numbers of these pairs apparently decrease to lowvalues. The changes in the relative number of the pairscoincidentally correspond to the changes in internal energy asplotted in Fig. 2, indicating that the atomic local structuralchanges result in energy changes during heating.

4. Conclusions

In this work, MD simulations described by J. Mei version of theEAM potential reveal local structure changes of the Cu201 clusterat elevated temperatures. The structural changes of this clusterclearly include a slowly increasing region of energy at a low-temperature range, where atoms keep their ordered positions, afluctuating energy increase range, where atoms interchange their

ARTICLE IN PRESS

L. Zhang et al. / Physica B 404 (2009) 205–209 209

positions in regions I–III, and finally a rapidly increasing regionwith a higher slope, where the surface region atoms move intonear-surface and inner regions of this cluster. At elevatedtemperatures, initial structural changes present atom movementsfrom the inner region into the outer regions, accompanyingtransformations of the local atom packing in the three regionsfrom FCC to HCP structures as well as some icosahedronstructures. Atoms interchange their positions in the near-surfaceand surface regions, and finally the surface atoms move into thesub-surface regions, resulting in the overall melting of the wholecluster. As a result of atom movements with increasing tempera-ture, volume expansion and local multi-structures of this clustercan be observed in its melting process. The analysis as notedabove indicates that the melting scenario revealed in oursimulations is different from the structural evolution of thesesmaller or larger clusters, involving surface melting or thickeningof a liquid layer. In general, owing to the correlated motion ofatoms, the melting process of the studied cluster involves the localstructural transformation from a low-temperature optimal geo-metry to the local structural coexistence of the FCC, HCP, andicosahedral structures followed by the atoms becoming graduallydisordered in the solid–liquid state.

Acknowledgement

We acknowledge the financial supports from the NationalNatural Science Foundation of China (Grant no. 50572013) and the

National Basic Research Program of China (Grant no.G2006CB605103).

References

[1] W. Eberhardt, Surf. Sci. 500 (2002) 242.[2] F. Baletto, R. Ferrando, Rev. Mod. Phys. 77 (2005) 371.[3] R.E. Kunz, P. Blaudeck, K.H. Hoffmann, J. Chem. Phys. 108 (1998) 2576.[4] D. Gotwald, G. Kahl, C.N. Likos, J. Chem. Phys. 122 (2005) 204503.[5] M. Howe James, Interfaces in Materials, Wiley, New York, 1997, p. 116.[6] M. Kabir, A. Mookerjee, A.K. Bhattacharya, Phys. Rev. A 69 (2004) 043203.[7] Z.H. Jin, H.W. Shen, K. Lu, Phys. Rev. B 60 (1999) 141.[8] V.G. Grigoryan, M. Springborg, Phys. Rev. B 70 (2004) 205415.[9] K. Jug, B. Zimmermann, P. Calaminici, A.M. Koster, J. Chem. Phys. 116 (2002)

4497.[10] E.M. Fernandez, J.M. Soler, I.L. Garzon, L.C. Balbas, Phys. Rev. B 70 (2004)

165403.[11] C.W. Bauschlicher Jr., S.R. Langhoff, H. Partridge, J. Chem. Phys. 91 (1998)

2412.[12] W.Y. Hu, S.F. Xiao, J.Y. Yang, Z. Zhang, Eur. Phys. J. B 45 (2005) 547.[13] L. Zhang, C.B. Zhang, Y. Qi, Phys. Lett. A 372 (2008) 2874.[14] H. Lei, J. Phys.: Condens. Matter 13 (2001) 3023.[15] M.F. Rosu, F. Pleiter, L. Niesen, Phys. Rev. B 63 (2001) 165425.[16] T. Jacob, J. Anton, C. Sarpe-Tudoran, W.–D. Sepp, B. Fricke, T. Bastug, Surf. Sci.

536 (2003) 45.[17] G. Boisvert, L.J. Lewis, Phys. Rev. B. 56 (1997) 7643.[18] O.S. Trushin, P. Salo, T. Ala-Nissila, Phys. Rev. B 62 (2000) 1611.[19] S. OzC- elik, Z.B. Guvenc- , Surf. Sci. 532–535 (2003) 312.[20] Z. -P. Shin, Z. Zhang, A.K. Swan, J.F. Wendelken, Phys. Rev. Lett. 76 (1996)

4927.[21] R.C. Longo, C. Rey, L.J. Gallego, Surf. Sci. Lett. 457 (2000) L441.[22] J. Mei, J.W. Davenport, G.W. Fernado, Phys. Rev. B 43 (1991) 4653.[23] L. Zhang, S.Q. Wang, H.Q. Ye, Chin. Phys. 15 (2006) 610.[24] J.D. Honeycutt, H.C. Andersen, J. Phys. Chem. 91 (1987) 4950.[25] A.S. Clarke, H. Jonsson, Phys. Rev. E 47 (1993) 3975.