Formation of Superclusters from Metallic Clusters

(1)

Zeitschrift fürPhysikalischeChemie, Bd. 199, S. 87-98(1997)

Formation of

_{Superclusters}

from Metallic Clusters

By

Hans-Gerhard

_Fritsche1,

Hans Müller1 and

_Benjamin

Fehrensen2

1 _{Institut für}

Physikalische

ChemiederChemisch-Geowissenschaftlichen Fakultät derFriedrich-Schiller-Universität, AmSteiger3,D-07743 Jena,Germany

2 _Zürich,_{Institut für}_{Physikalische}

Chemie, Zentrum, Universitätsstraße22, CH-8092Zürich, Switzerland

(ReceivedAugust 23, 1996)

Mesoscopic

particles

I

_Supercluster

I

_Size

_effects

I

Surface

dependent

properties

I Embeddedatom

_{approximation}

Schmidetal. [3] reported on the

possibility

to build upexperimentally microcrystalline

modificationsofmetals with icosahedral clusters M„ asthebasic units of the structure.

Wetheoretically investigatetwo

_{possibilities}

to constructsuch

_{superclusters}

: a) a forma-tionof shell-like superclusters, andb)agenerationofsuperclustersfromsuperclusters.

Thenumericallyobtained results_{agree very}_{well with many}

_experimental

dataofthe

newmetallic modification.They explainthegrowthofsuperclusters

by

asimple

analyti-cal model. Size effects ofsome mechanical propertiesofsuch superclusterscanbe

pre-dictedby

analytical

relations.

Schmidetal. _[3] berichtetenvonderexperimentellen Erzeugungmikrokristalliner Modi-fikationenvonMetallen mit ikosaedrischen ClusternMaals strukturellenGrundeinheiten.

Wir untersuchten daraufhin mit theoretischen Methoden zwei Wege der Konstruktion solcher Supercluster: a) die Bildung schalenförmiger Supercluster und b) die Bildung

vonSuperclusternausSuperclustern.

Die numerisch gewonnenenErgebnissestimmen rechtgutmitexperimentellenDaten derneuenmetallischenModifikation überein.AufihrerGrundlage ergibtsich ein

analyti-sches Modell des Wachstums von Superclustern, welches die Vorhersage der

Größen-abhängigkeit mechanischerEigenschaftenvonSuperclusternerlaubt.

1. Introduction

The

_high

_stability

of transition metal cluster

_compounds

with a core of

geometrically

closed shells of metal atoms is wellknown

[1, 2].

Such

com-pounds

can be used to

produce ligand-free

ML1

clusters

by degradation

of

(2)

88 Hans-GerhardFritsche, Hans Müller and _BenjaminFehrensen

M13

clusters do not

_decompose

into smaller

_particles

_(as

_supposed

_[4]),

but

they

aggregate

to

superclusters.

Two lines of the formation of

_{superclusters}

have been found

_{experimentally}

:

1)

the

_generation

of

_{superclusters}

of one or two full shells of clusters

M13

[5, 6]

13

_M13

->

(M13),3

(a)

55

M13

->

_(M13)55

(b)

(1)

and

2)

the

_generation

of

_{superclusters}

from

_{clusters/superclusters}

_[3,

6,

9]

13

M13

->

(Mt3)13

(a)

13

_(M13)13

->

((M13)13),3

(b)

13

_((M13)13)13

->

_{((( 13)13) 3) 3}

(c)

(2)

Here,

only

the first

_steps

of both lines are identical.

Some years ago we derived from a

simple

jellium

model within an iterative version this

_{aufbauprinciple}

as a

disign principle

for new metals

[7, 8],

The formation of

_{superclusters}

from

_{superclusters}

can _be

repeated

step-wise

_(Fig.

1).

The

_{resulting microcrystalline products}

of the metals

Au, Co,

Pt,

Rh,

and Ru have been

_structurally

analized

_by

various

_experimental

techniques

(time

of

_{flight secondary}

ion mass

_{spectrometry,}

X-ray powder

diffraction,

Mössbauer

_{spectrometry).}

It has been

found,

that the new

me-tallic modification is stableat low _{temperatures.}At

_higher

_temperatures

_(for

((AU|3),3)13

at700

—

800

_K),

it is transferredinto the structure of normal bulk

metal

_[9].

It can be

expected,

that this new

microcrystalline

modification has

properties

different from that ones of the

corresponding

bulk metal. Some ofthemwill be

_investigated

in_{this paper}

_{theoretically by}

useof the Embed-ded Atom Method

_(EAM).

2. The

embedded

atom

method

_(EAM)

The embedded atom method has been described in detail in many papers

[10—13].

The method was

successfully

applied

to _many

_problems

of metals

(clusters

and

_bulk).

For a

review,

see

[13].

Here,

we

only

_report

on some few

_aspects

of the method.

We start with an

approximation

of the total

potential

_{energy of}

interac-tion

_by

additive attractive

_{embedding energies,}

_Fh

and

_pairwise

additive

energies

of

_repulsion,

of the atoms of the cluster

_MN\

=

F,

( ,,,)

+

„

_{( )}

·

(3)

Formation of_{Superclusters}from Metallic Clusters 89

Fig.1.

_Supercluster

((M,,),,),,:generation line2, productofEq.(2b).

In this

_expression,

_{,,, is the host electron}

_density

at atom i due to the

remaining

atoms,

_{/\( )}

is the energy to embed atom i into the

background

electron

_density

, and

_(7?,,)

is the core-core

pair repulsion

between

atoms i and

_j

_{separated by}

the distance

_/?„.

The host _electron

_density

_is

approximated by

the

_{superposition}

of densities of valenceelectrons

Q„,=

_Q'iRg),

(4)

where

_'

_(Ru)

is the electron

_density

at atom i contributed

_by

atom

_j.

The atomic densities will be taken from the tables of Clementi etal.

[14],

and McLean et al.

[15].

Values of the functions F and can be calculated from the formal definitions of these

_quantities

within the

_{density-functional}

framework

_[16],

but it is usualtoestablishF and

_empirically

from

physi-cal

_properties

of solids

(lattice

constant, elastic constants, cohesive energy, and

_{vacancy-formation}

_energy)

[10, 12].

Numerical calculations have been carried out here for M =

Li, Ni,

and

Au. We used the

_{EAM-parametrization}

of

_Zhang

etal.

(lithium, [17]),

and of Voteretal.

_(nickel,

_[18]).

For

_gold,

wedeterminednew

EAM-parameters

(4)

90 Hans-GerhardFritsche, Hans Müller and _BenjaminFehrensen Table 1. Equilibrium dataof icosahedral clusters M,, (M = Li, _{Ni, Au)} optimized by

EAM calculations: r, = _{shortest interatomic}

distance, E/13 = binding

energy peratom.

rm = cut-off radius of EAM-interactions.

M,3 rc„,

[Á]

r,

[Á]

-E/13 [eV] rcJr,

Li„ 4.9629 2.823 1.279 1.758

Ni„ 4.7895 2.382 3.034 2.011

Au,, 3.4739 2.630 3.101 1.321

The contributions to the energy of interaction of atoms

separated

from

each other morethan adistancercut are_{very small}

(contributions

tothe host electron

_density

_,,., as well as contributions to the core-core

repulsion

).

Usually, they

are

neglected.

(Their

neglection

is

incorporated

into the deter-mination of the

_{EAM-parameters}

_by

the

_adaption

of

_experimental

data of

solids.)

3. Results

3.1 Icosahedral basic clusters

M13

First,

we

optimized

the basic icosahedral clusters

MI3.

The resultsare

given

in Table 1. The shortest interatomic

distances,

ru and the

binding energies

per atom,

£713,

are

qualitatively

correct

(between

the

_experimental

values

of bulk and thatones of the diatomic

molecule).

For detaileddiscussion, see

[19].

The last column of Table 1 shows the maximum _range ofinteratomic interactions in units of the shortest interatomic

distance,

r,, of the

icosa-hedral.

3.2 The dimeric

_supercluster

(M13)2

It has been

verified,

that two icosahedral clusters

_M,3

at their

_equilibrium

distance,

r2, have a

rigid

structure to a

high degree

of accuracy

[20].

As an

example, Fig.

2 shows the cluster-cluster interaction energy,

AE(L

=

2)

=

E(2

·

13)

-2

£(13),

ofthe dimeric

_supercluster

(Au,3)2

in

dependence

onthe distance between the both

clusters,

r2. The values at the

equilibrium

are: = -0.45

eV,

r2 = 7.89 A. Inharmonic

approximation,

the

_frequency

of the relative oscillation of the both clusters is =

3.8 · 1011

Hz,

the

corresponding

wave

length

is = 0.08 _cm.

3.3 Formation of

_{bigger superclusters}

Now,

we

study

the formation of

superclusters

(M13)L

from L

rigid

(5)

In

fact,

the energy of the cluster-cluster interaction

= E

(L

13)

-L

_£(13)

₍₅₎

will

_depend

on the relative orientation of the

clusters,

but it will be shown in section

3.3.1,

that

_changes

of this orientation will influence the

_quantity

_{only by}

a constant factor.

_{Interesting qualitative}

results canbe obtained

by

the

_{only optimization}

of the distance r2 between two

neighbouring

clus-ters of the

supercluster,

too. In this sense, we define here the structure of the

_supercluster

tobe fixed. All clusters

_M,3

inside of the

_supercluster

have the same orientation oftheir local z-axis.

3.3.1 Generation line 1

As an

_example

of the

generation

line

1,

we

_report

on the formation of icosahedral

_{superclusters QJ.i3)L consisting}

of closed icosahedral shells of icosahedral cluster units

Li13.

Investigations

of other structures _of

super-clusters

(linear,

squared, simple

cubic,

body-centered

cubic,

face-centered

cubic)

gave similar results. In this

notation,

the centres of the basic clusters

Li,,

define the

_polyhedron

of the

_{supercluster.}

We studied

_{superclusters}

of =

1,2,...,8

shells with L =

13,55,...,2057

cluster units

or =

169,715,...,26741

atoms and determined the

optimum

shortest distance r2

oftwo

neighbouring

clusters

Li,

,for each

supercluster.

The resultsare

given

(6)

92 Hans-GerhardFritsche.Hans Müller and BenjaminFehrensen Table2. Equilibrium data of shell-like icosahedral superclusters (Li13)¿ (generation

line1):r2 = _{shortest intercluster}pairdistance, /L =

cluster-cluster interaction energy per cluster unit, AE/(L CN,) = interaction energy of_one

pair

of clusters of the super-cluster.

L _r2

[k]

-AE/L [eV] -AEI(L CN,) [eV]

1 13 8.3199 0.3906 0.1209 2 55 8.3814 0.4969 0.1168 3 147 8.3969 0.5458 0.1153 4 309 8.4057 0.5737 0.1145 5 561 8.4113 0.5916 0.1140 6 923 8.4152 0.6040 0.1137 7 1415 8.4181 0.6132 0.1135 8 2057 8.4203 0.6203 0.1134

Line 1 :

Supercluster

(Li,3)L

1-0\.

^ _0.8

on\ I.I.I

0.00 0.10 0.20 0.30 0.40 0.50

L-"

Fig.3. Sizeeffect ofthecohesive_energy,zl£/L, of icosahedral

_{superclusters}

(Li,,),,with

= 1,2,...,8closed shells(generation line 1).

We see:

1)

The shortest interclusterdistance, r2, is

nearly

constant.

2)

The cluster-cluster interaction energy per

cluster, /L,

increases with the size of the

_{supercluster. Fig.}

3

_shows,

that the size-effect of /L can be described

by

the

equation

/L«_a +

b

·Lrm

.

(6)

This relation is a first indication ofthe

fact,

that there is no on

principle

a difference between size effects of

_{superclusters}

and size effectsof clusters.

However,

an

important quantitative

difference exists: The distances between

(7)

Formation ofSuperclustersfrom Metallic Clusters 93

thecentresoftwo

_neighbouring

clusters in the

_supercluster

aremuch_greater

than the distances between two

_neighbouring

atoms in the

corresponding

clusters. Due to

_it,

in

_{superclusters}

not

_only

the

_{cluster-pair approximation}

of the cluster-cluster interactionenergy

»

,

(7)

/< j

is

_justified,

but also the restriction to the

_only

contributions of the nearest

neighbouring

pairs

of clusters:

r>

¡EÁRÚ

for

Ru=r2

EU{RU)

=

\

(7 a)

L

0 otherwise

(R,,

= distance between the _centres of the cluster / and

7,

r2 = _nearest

neighbouring

distance oftwo

clusters).

3)

To

_verify

the relations

(7)

and

_{(7 a),}

we now define a mean first coordination

number,

C/V,(L),

of a cluster

Li,,

inside of the

supercluster

(Li13)¿

as the total number of all nearest

_neighbouring

clusters devided

_by

number L.

Here,

for the sake of

_simplicity,

we _{suppose the} nearest

periph-eral and the nearestradial distances between two clusters

Li13

of the

icosa-hedral

_supercluster

to be the same.

The last column of Table 2

shows,

that the cluster-cluster interaction energy of one

pair

of

interacting

clusters, E,

=

AE/(L

C7V,(L)),

is

nearly

constant for all icosahedral

_{superclusters.}

That means: the formation of

superclusters

is

_simply

controlled

_by

an

optimization

of the first coordi-nation of the cluster units inside of the

_{supercluster:}

AE/L « 0.5

E,

CNAL)

.

(8)

Itis

obvious,

that

_equation

(8)

is

_equivalent

with the

_equations

₍₇₎

and

(7 a).

The

_validity

of the relations

₍₆₎

and

₍₈₎

has been

_proved

for various

structures of

_{superclusters}

with different relative orientations of the

icosa-hedral units

Li13:

linear

chains,

squared

structures,

_simple

_cubes,

body-cen-tered cubes, and face-centered cubes. In _every _case, the

_{description by}

the

simple analytical

relations

₍₆₎

and

₍₈₎

was excellent.

Therefore,

we con-clude:

The size effect of the cluster-cluster _{interaction energy in}

_{superclusters}

is

_mainly

a size effect of the

averaged

first coordination number of the cluster units. The chemical nature of the

specific

metal,

the internal struc-ture of the cluster units aswell as the relative distances and orientations of

the clusters inside of the

_{supercluster only}

determine the values of the

con-stants a, b, and

£,

in the

Eqs.

(6)

and

(8).

This result

_{explains, why}

mass

_spectra

of

superclusters

(Au,3)L

just

show

the

_preferred

occurrence of

superclusters

with L = 13 and L = 55 cluster

units

_[5,

_6]:

The

_{corresponding}

structure of a full shell icosahedron

(8)

94 Hans-GerhardFritsche,HansMüller and_BenjaminFehrensen

Ni,3

_

(Ni„)„

r,

[Â]

r4

[Á]

Fig.4. Formation energies of icosahedral superclusters ofnickel, _AEk, for 4 steps of

generation

line 2.

3.3.2 Generation line 2

Asa second

possibility,

we

investigate

now a

stepwise generation

of icosa-hedral structures of

_{clusters/superclusters}

of the 3 elements

Li, Ni,

and Au:

step 1: 13M—»

M13

(see

section

3.1.)

r, =

optimum

distance ofM—M

step

2: 13

M13

—»

(M13),3

(see

section 3.3.1 for =

1)

r2 =

optimum

distance of

M,3—M,3

step

3: 13

(M13),3

->

((M,3)13)13

r3 =

optimum

distance of

(M13)13—(A/13),3

step

4: 13

_((M13)13)13

->

_{(((M13)13)13)13}

r4 =

optimum

distance of

((M,3),3)13-((M13),3)13.

Each

_step,

k,

uses the

optimized product

of the

preceding

_step,

k

—

1,

as a

rigid

unit to _{build up the} next icosahedral structure with a new op-timized cluster-cluster distance rk.

(9)

Table 3. Equilibrium data of k = 1, 2, 3, 4 steps ofgeneration ofsuperclusters from

superclusters ofNil3 (line 2): rk = _shortest supercluster-supercluster

distance, AEk =

binding

energyofsuperclustersper

supercluster-unit.

k rk

[À]

-AEk [eV]

1 2.332 3.156

2 6.764 1.662

3 20.514 0.636

4 61.540 0.636

Fig.

4

_presents

the

_{cluster/supercluster—formation energies}

of line 2

AEk

=

£~^£'

for * =

1,

2, 3,4

,

(9)

in

_dependence

of the distance rk for the element

nickel,

Table3 the

corre-sponding

values at the

equilibrium.

Qualitatively,

the results of lithium and

_gold

are similar. Ofcourse, the

magnitudes

of

_AEk

andrk differ from thatones of nickel.

Acareful

_analysis

of the structures of the

_{superclusters}

showsa

shorten-ing

of the distances between the outer

contacting

atoms of the

_interacting

units of the

_{superclusters}

in

_comparison

with the constant value of r,:

-0.232

Á

in

_{step 2,}

-0.010

Á

in

_{step 3,}

and -0.002

Á

in

_step

4. In the same _{sequence, the relative values of the cluster-cluster-formation energy,}

AEkIAEu

decreases: 0.526 in

_{step 2,}

0.201 in

_{step 3,}

and0.201 in

_step

4.

Obviously,

the

_generation

line 2 does not

_stop

at

_step

4. The

_growth

of

superclusters

can be

repeated stepwise

with a _constant

_energetic

_rate of

growth.

4. _Properties

of shell-like

_{superclusters}

_(Mi3)L

4.1 Cohesiveenergy of

_{superclusters}

Now,

wediscuss the

generation

line 1 more

generally

_andassume the

super-cluster

_(Afi3)¿

to have a structure of

_{geometrically}

closed shells of units

M13. Then,

the number Lof cluster units canbe

expressed by

acubic

poly-nomial of the number of

shells,

n :

L{n)

=

a3n3

+

_a2n2

+ a,n + a0.

(10)

The coefficients ak of

Eq. (10)

are

given

in

[21]

for 15

highly

symmetrical

particles.

The total energy of interaction between all units

M13

of the supereluster is

AE = 0.5 ·L

(10)

96 Hans-Gerhard Fritsche,Hans Müller and BenjaminFehrensen

For shell-like

_{superclusters,}

also the mean

L-dependent

coordination

num-bers

_CW,(L)

are

analytical

expressions

ofthe number of

_shells,

n:

CN,(L(n))

=

2(b3n3

₊

b2n2

₊

btn

₊

b0)IL.

(12)

The values of the coefficients

b¡

are

given

in

[21].

Thus,

from the

Eqs.

(11)

and

(12)

follows a

polynomial representation

ofthe total interaction energy of the

_supercluster

AE(n)

=

Et

3 3

₊

b2n2

₊

b,n

₊

bn).

(13)

From the

_Eqs.

(10)

and

(13),

the cohesive energy of the

supercluster,

/L,

can be written as

/L =

A0

+

A_,/?-' +A_2n-2

₊

...

(14)

with the bulk-value of the

_supercluster

{AE/L\

=

A0

= ,

(15)

Ű3 and

-1

=

(b2a3

—

)

E,/(aj),...

4.2 Surface

_{dependent properties}

of

_{superclusters}

We _{define the surface energy,}

_ES(L),

ofa

_supercluster

_ofL

rigid interacting

basic units

_M,3

_by

the relation

ES{L)

=

AE(œ)

-AE(L)

.

(16)

Putting

the

_Eqs.

_(10,

14, and

₁₅₎

into

_Eq.

_(16),

we obtain

Es(n)

=

-[B2n2

+

,

₊

B0

₊

^, '1

₊

···],

(17)

with

B2

=

a3A-u

ß,

=

a3A-2

₊

a2A^„

...

The

_question,

what is the surface ofa

supercluster,

is difficult

quantitatively

to be answered. As a crude

approximation,

we assume the surface of the

supercluster

tobe the surface ofthe

_polyhedron

withfaces

_simply

construct-ed from the centres of the cluster units

_M13

at the outer_{shell of the}

super-cluster.

_(In

_reality,

the surface must be

greater.)

Then,

the surface area,

A,

follows as

A=fr2,

(18)

with r

being

an

edge

length

of the

polyhedron.

The

quantity /

is a form factor

_specific

for the

_{special polyhedron.}

At an

edge

ofa

supercluster

of

shells,

(n

+

₁₎

cluster units

Mn

are

positioned

with the distance _r2between

two

_neighbouring

units.

Thus,

the

_{edge length}

can _be

expressed by

the shell number

(11)

Formation ofSuperclustersfrom Metallic Clusters 97

and the surface area is a

squared polynomial

ofn:

A

_=fr\

_(n2

+ In +

_1).

₍₂₀₎

Now,

a

simple

relation of the

specific

surface energy, _, follows:

with_c0 =

( )

=

B2/(frl),

c_, =

(£?,

-2

_B2)/(fr22)

Similarly, analytical

formulae for other mechanical

_properties

_of super-clusters like the

_isotropie

surfacestressorthe

compressibility

canbe derived

[21, 22].

5. _Concluding

remarks

Icosahedral clusters

_M13

of alkaline

metals,

transition

_metals,

and noble

met-als have diameters

_greater

than or

equal

_to the range of the direct interac-tions between two metal atoms. Our numerical calculations have

_shown,

that such clusters interact in

_{superclusters}

_(M13)t

at the

_{equilibrium by}

the

only

attraction of the nearest

neighbouring

_pairs

ofthe clusters. Because of the collective character of the metallic interaction considered

_by

the non-linear

_dependence

of the

_embedding

functions Fon the host electron

den-sity

(Eqs.

3 and

_4),

this result is not trivial.

Fora defined structure of the

_{supercluster,}

which has been determined

experimentally

[1,

3, 5, 6,

9],

the cohesive energy of the

supercluster

only

depends

on the size

dependent

mean coordination ofa cluster

by

the next

clusters inside of the

_{supercluster.}

This result

_justifies

a

simple

topological

model of the cluster-cluster interaction in metallic

_{superclusters proposed}

eight

years ago

[23].

Acknowledgement

The authors are

indepted

to the Deutsche

_{Forschungsgemeinschaft}

_and to

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