Z. Phys. D - Atoms, Molecules and Clusters 20, 239-242 (1991) Atoms, Molecules

and Clusters ~Phy,~k D

© Springer-Verlag 1991

Structural and dynamical properties of transition J. Jellinek and I.L. Garz6n**

Chemistry Division, Argonne National Laboratory, Argonne, IL 60439, USA

Received 10 September 1990

metal clusters*

Abstract. Results of molecular dynamics simulation stud- ies of structural and dynamical properties of 12-, 13-, and 14-atom transition metal clusters are presented. The cal- culations are carried out using a Gupta-like potential expressed in reduced units. The transformation to abso- lute units involves two size-dependent parameters which effectively convert the potential into a size-dependent one. The minimum energy geometries of the clusters are ob- tained through the technique of simulated thermal quen- ching. A melting-like transition is observed as the energy of the clusters is increased. A novel element of the trans- ition is that it may involve a premelting state.

PACS: 36.40

I. Introduction

Structural and dynamical properties of atomic and molecular clusters are a subject of increasingly intense research activity theoretically and experimentally. The interest in clusters is fueled by the important, in a sense unique, role these systems play from both, fundamental- cognitive and applied-technological, points of view. The changes in the properties of clusters with their composi- tion (material) and size make them ideal "microscopic laboratories" for studying the different forces responsible for the cohesion between atoms and molecules. Under- standing of the size-dependence of cluster properties can lead to a better control over a variety of technologically important processes.

While appreciable progress has been made in the last few years in theoretical studies of van der Waals, hydro-

* Work supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Chemical Sciences, under Con- tract W-31-109-Eng-38. I.L.G. was partially supported by Project DGAPA-UNAM (IN-10-42-89) ** Permanent address: Instituto de Fisica, Universidad Nacional Aut6noma de M6xico, 22800 Ensenada, Baja California, M6xico

gen-bonded and, to somewhat lesser extent, covalent clus- ters (see, e.g., [ 1-1 and citations therein), relatively little has been derived concerning structural and dynamical proper- ties of metal clusters. This is especially true for transition metal clusters. The main hinderance to theoretical dynam- ical studies of metal clusters is in that no truly adequate potentials are known for these systems at present. The few dynamical studies of transition metal clusters [2-4] used semiempirical potentials with most, if not all, the param- eters fitted to bulk properties.

A crucial criterion of the adequacy of a potential is its ability to mimic the size-dependence of cluster properties. Since the potentials used for metal clusters at present are, in fact, models, one may attempt to build the size- dependence into a model potential explicitly. This can be achieved by making some of the parameters of the poten- tial size-dependent. In addition to the heuristic argument in favor of size-dependent model potentials one can in- voke also a conceptual one. It is well-appreciated that many-body effects play an important role in metallic cohesion. Introduction of size-dependent parameters into a potential may be a way to effectively account for the individually unknown two-body, three-body, etc. inter- actions; cf. [5]. As models, the size-dependent potentials should be simple and efficient in large-scale numerical simulations.

In the next section we introduce a Gupta-like size- dependent potential to mimic transition metal clusters and outline the computational procedure used. The nu- merical results and their discussion are presented in Sect. III. A brief summary is given in Sect. IV.

II. The potential and the computational procedure

Recently Sawada and Sugano [3] used the potential

= 2- j~l i= 1 \ro

- - [ j=~ exp ( - 2q(~oS- l ) ) l ~/u } (1)

240

to model the melting transition in 6- and 7-atom trans- ition metal clusters. This potential derives from Gupta 's [6] expression for the bulk cohesive energy which is based in tight-binding considerations and which has been used to describe surface relaxation in transition metals. In (1) r~ is the distance between the i-th and the j-th atoms, and U, A, ro, p and q are adjustable parameters. It has been suggested [7] that the values p = 9, q = 3 are appropriate for transition metals (cf. [8]). The value of A can be determined by minimizing the cohesive energy at the known lattice structure of the bulk material (for this minimization r o is set equal to the nearest-neighbor dis- tance in that lattice).

We use the potential (1) but treat the remaining two parameters U and r 0 not only as material- but also as size- dependent, which is indicated in what follows by attaching the subscript n to them. The explicit size-dependence can, however, be formally eliminated from the potential by rewriting it in reduced units V* = V / U , and r* = ru/ro,: ln{

= exp ( - p(r* -- 1)) V* 2~--~1 A

- [ ~ exp ( - 2 q ( r * - 1))] 1/2 } (2)

(the remaining dependence on size enters through the summations). The value of A = 0.101036 (cf. [3]) was determined by minimizing the cohesive energy of an fcc crystal; to achieve a six significant digits convergence a spherical slab of 429 atoms was needed. Employing the reduced-form potential (2) we carried out simulation stud- ies of the structure and dynamics of 12-, 13-, and 14-atom clusters. Newton's equations of motion were solved using the Verlet algorithm [9]. The time step of 7.8' 10-16S assured conservation of the total energy of the cluster within 0.5%. The initial values of the coordinates and momenta were chosen to yield nontranslating and non- rotating clusters.

III. Numerical results and discussion

Using the technique of thermal quenching (i.e. repeatedly setting the momenta of the particles to zero) we deter- mined minimum energy structures of the clusters. A clus- ter was deemed quenched when its residual time-averaged kinetic energy per particle in reduced units dropped below the value of 10 -6. The n = 13 cluster is most stable in icosahedral geometry with the energy V*o = -12 .463. The hcp and cubooctahedral structures are almost degen-

-- Vcubo erate energetically (V*cp 12.240, * = - 12.238). The most stable configurations of n = 12 (V* = - 11.292) and n = t4 (V* = - 13.398) clusters are obtained from the icosahedral n = 13 by removing one surface atom and adding one a tom to a threefold surface site, respectively, and allowing for the relaxation of the systems.

To monitor the response of the clusters to the increase of their internal energy we plotted their short-time (500 steps) averaged kinetic energy per particle as a function of time for a wide range of fixed total energies per particle.

The short-time averaging is introduced to damp the fluc- tuations in the kinetic energy due to the vibrational motions. A trajectory of 10 5 steps was run for each total energy. The curves for different total energies (labeled as a, b, c, etc.) are shown in Fig. 1. At low total energies the short-time averages of the kinetic energy form narrow unimodal distributions. Over a finite range of intermedi- ate total energies the distributions for n = 12 and n = 13 branch into two or more "modes". At higher energies the widths of the modes increase and the individual branches

b-

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l-- <

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I-- tU z x,"

g b- ~< >-

z ¢2

ILl Z x~

.lo

.08

.06

.o4

.02

.oo

.02

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.04.

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.00

n=12 l

b

l]

0 200 400 600 800 tooo

0 260 460 660 ar0

e

d

C b 0

lOOO

TIME (7.8E-14 s)

Fig. 1. Short-time averaged kinetic energies per particle (in reduced units) as a function of time. The cases correspond to the following total energies per particle (in reduced units): n = 12: a) - 0.916, b) - 0.896, c) - 0.879, d) - 0.839, e) - 0.783; n = 13: a) - 0.941, b) - 0.915, c) - 0.869, d) - 0.809; n = 14: a) - 0.950, b) - 0.942, c) - 0.936, d) - 0.918, e) - 0.897, f) - 0.855, g) - 0.794

become indistinguishable. This is reminiscent of what has .18- been observed in studies of Lennard-Jones clusters [10], and is an indication of structural changes associated with a solid-to-liquid like transition. But the melting behavior of transition metal clusters contains also elements which < are new as compared to the melting features of the Lennard-Jones clusters. A detailed analysis will be pre- z"' sented elsewhere. Here we give only an illustration for the "'

(,3 case of n = 14. Examination of Fig. 1 clearly shows that this cluster "melts" in two stages. In the first, low energy stage the distribution of the short-time averaged kinetic " energy changes with the total energy from a narrow unimodal (graph a) to a well-resolvable bimodat (graphs b and c) and then back to a unimodal, but a broader one (graph d). In the second, high energy stage (graphs e, f and g) the 14-atom cluster behaves similarly to the 13-atom one. We have identified the first stage as resembling the so-called fluctuating state described by Sawada and ~_° Sugano [3] for a 6-atom transition metal cluster. Two topologically similar n = 14 isomers, one with the 14th

OJ atom in a three-fold site of the 13-atom icosahedron and z

W

the other with the 14th atom in a four-fold site of a locally o distorted 13-atom icosahedron, interconvert through a u~ collective breathing-like motion with a rate which de- z pends on the total energy. We label this stage by the term "~ "premetting". In the second stage the premelted state coexists with other, topologically different n = 14 isomers and eventually melts (in the same sense as Lennard-Jones clusters do [10]) as the energy of the cluster is increased. Premelting is a phenomenon not found in Lennard-Jones clusters. =

Figure 2 displays the long-time (over the entire runs) 2 <

averages of the kinetic energy per particle vs total energy per particle (caloric curves). One notices the correlation between the changes in the slope of the caloric curves and ~ z the corresponding changes in the patterns of the short- c) time averaged kinetic energy (Fig. 1). The solid-to-liquid transition region for clusters can be defined as the range(s) z

5£ of the total energy between the points at which the corresponding caloric curves change their slope. Note that there are two transition regions for n = 14. The one at lower energies corresponds to premelting. The abrupt changes in the temperature (T) dependence of the root- mean-square bond length fluctuations (6) displayed in Fig. 3 corroborate the interpretation of the nature of the transition as solid-to-liquid like. T and 5 are defined by

k T - 2 ( E k ) 2 i £ ((rZ~) - - (rij)2)l/2 3 n ~ ' 6 - n(n ~ 1) '< i ~-ru) '

O) where E k is the total kinetic energy of the atoms in the cluster, k is the Boltzmann constant, and ( ) denotes the long-time average. The 5 graph for n = 14 displays clearly the premelting (first abrupt change in 5) and the melting (second rapid change in 5) stages of the transition.

The results presented above are expressed in terms of reduced energy. It is the reduced-form potential, (2), with parameters depending only on the material, which defines the qualitative features of the structure and dynamics of clusters. In order to convert the energetic and structural

2 4 1

14" n=12

.11"

o

.07- o ° v

e , °

.04- / "

• -.g2 4 4 18-

n=13

.11

.07- ." , , , j t

# .04-

.%-/ , -.92 -.84

.18-

o

a

o

o

o

o

o

o

o

o

o

-.60

-.60

14" n=14

.11- o

.07. ." o"

.04" ~/o°°"

'0-t) -.92 -.84 48 -60 TOTAL ENERGY/ATOM

Fig. 2. Caloric curves. The full circles denote the solid-to-liquid transition region. The low-energy transition region for n = 14 (indi- cated by an arrow) corresponds to the premelting stage

data into absolute units one has to know the values of U, and ro,. These are the parameters which introduce the size-dependence into the potential, (1). The character of the size-dependence in this particular implementation is, thus, of a scaling nature: the energy- and length-scales depend on the cluster material and size. The values of U, and ro, can be determined if, for example, the energy and configuration of a stable structure of the n-atom cluster of interest are known (e.g. from experiment or ab initio calculations). No such information is available for tran- sition metal clusters at present. In order to gain a quanti- tative estimate of the "melting temperature" of Nil3 we fitted the binding energy of the icosahedral n = 13 cluster to

242

Z O H F- <:

I-" r j

d LL

"r p- CO Z W J

0 Z 0 m

03

rr

Z O H I-- <

t"- C.) :D / LL

I I"-- r._9 Z IjJ _1

O Z O m

0 3

0E

Z O H

< D t-- f_3

d LL

I-- (.9 Z Ld

£3 Z O c o

O3 2r r r

.40 ¸

.32 ¸ o

oo °o

o°

24.

.16

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00060 .o 2 .o 4 .40-

o o

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n . - 1 2

.096 .128

.32

24-

.16-

.08"

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n=13

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I

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I

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°

~0

oo

o

,o°°° ° n= 14 °.'°°°

° .

.160

.160

.032 ' ......... .064 .o96 558 .t6o

kT

Fig. 3. Root-mean-square bond length fluctuations as a function of temperature (in reduced units of kT); see text

the value calculated [4] using a version of the embedded- atom approach [11]. The value of U13 = 3.299 eV, ob- tained from this fit, puts the absolute melting temperature of Nil3 (specified here for convenience as the temperature at which 6 = 0.1845) at 1625 K. This is only slightly lower than the nickel bulk melting temperature (1728 K) in clear difference to the trend of drastic reduction in the melting temperature found for Lennard-Jones clusters [10]; the difference in trend remains for any selection of the melting point in the transition region. It is gratifying that the same size-dependent potential, when used with parameters cor- responding to gold, does lead to an appreciable reduction in the melting temperature for clusters [12]; such a reduc- tion has been observed in experiments on small gold

particles (see citations in [12]). Finally, we note that altering the values of the parameters to p = 10, q = 2.7, as suggested for nickel by Rosato et al. [8], does not change qualitatively the results presented here.

IV. Summary

A preliminary account of the results of a simulation study of structural and dynamical properties of transition metal clusters based on a model size-dependent potential is given. The dominant mechanism of a melting-like trans- ition taking place in these clusters, as their energy is increased, is similar to the one identified in Lennard-Jones systems: they evolve from solid-like to liquid-like through a "coexistence" (isomerization) stage. A novel element of the transition, found in the 14-atom cluster, is the so- called premelting state. The calculations indicate that the melting temperatures of nickel clusters may not be as different from the bulk value as those, for example, of gold clusters.

Model size-dependent potentials (there may be imple- mentations other than the one considered here) offer an additional flexibility in making the mimicking of particu- lar systems of interest more adequate. They may be particularly beneficial in dynamical studies of metal clus- ters.

References

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6. Gupta, R.P.: Phys. Rev. B23, 6265 (1981) 7. Ducastelle, F.: J. Phys. (Paris) 31, 1055 (1970) 8. Rosato, V, Guillope, M., Legrand, B.: Philos. Mag. A59, 321

(1989) 9. Verlet, L: Phys. Rev. 159, 98 (1967)

10. Jellinek, J., Beck, T.L., Berry, R.S.: J. Chem. Phys. 84, 2783 (1986); Beck, T.L., Jellinek, J., Berry, R.S.: ibid. 87, 545 (1987); Beck, T.L, Berry, R.S.: ibid. 88, 3910 (1988)

11. Voter, A.F., Chen, S.P.: Mater. Res. Soc. Symp. Proc. 82, 175 (1987)

12. Garz6n, LL., Jellinek, J.: Z. Phys. D-Atoms, Molecules and Clusters (this issue)