MICROCLUSTER IITHE GEOMETRIES OF SOFT-SPHERE PACKINGS

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MICROCLUSTER IITHE GEOMETRIES OF

SOFT-SPHERE PACKINGS

J. Barker

To cite this version:

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THE GEOMETRIES OF SOFT-SPHERE PACKINGS

J. A. BARKER

Department of Physics, Bedford College, Regent's Park, London NW1 4NS, England

Résumé. — Nous décrivons ici les résultats de nos essais de représentation d'agrégats non cristallins, formés d'empilements denses et comportant jusqu'à plusieurs centaines de sphères

molles. Nous montrons que des motifs à symétrie dodécahédrique peuvent être assemblés pour

former de grandes structures centro-symétrique basées sur le réseau radial 56.

Ces représentations d'agrégats sont alors utilisées, en même temps que des modèles relaxés de groupements aléatoires de sphères pour montrer que de grands domaines ordonnés peuvent exister dans des matériaux amorphes.

Abstract. — Described here are the results of our attempts to design densely-packed, non-crystalline clusters of up to several hundred soft spheres. It is shown that units of dodecahedral symmetry can be built into large centrally-symmetric structures based on the 56 radial net.

These designed clusters, along with relaxed models of random assemblies of spheres, are then used to demonstrate that large ordered regions may exist in amorphous materials.

1. Introduction. — Experimentalists working with small aggregates of atoms at very low temperatures (for example, clusters formed in supersonic nozzle beams [1]), continually have to ask themselves what geometric configurations these aggregates are assuming under the prevailing expe-rimental conditions. Often the conditions are not well known or are extremely complex, so neither nucleation theory (which, in any case, is well known for its inability to explain the early stages of cluster growth) nor computer simulations are able to make useful predictions. This being the situation, experimentalists are forced to compare their obser-vations with those calculated from a variety of models of atomic clusters.

The prime purpose of this paper is to describe and extend our knowledge of the important configu-rations of monatomic systems of up to about

1 000 atoms. Particular emphasis is laid on the geometries which, as a result of allowing the systems to be described in terms of assemblies of compressible (soft) rather than hard spheres, become not only possible but highly probable.

The second purpose of the paper is to elucidate some of the interrelations between the geometries of these soft-sphere packings and those of what would normally be thought of as random or amorphous assemblies.

2. Cluster models. — In table I are listed the main types of cluster that either have been used or could be used to model small monatomic systems at low temperatures.

For clusters of less than about thirteen atoms, it is not unreasonable to enumerate (using the

compu-T A B L E I

CLUSTER MODELS (Monatomic)

7V:sl3 Isomer Enumeration.

N " s l 3 Important Regions of Configuration Space CRYSTALLINE (f.c.c.) cube tetrahedron octahedron cuboctahedron spherical2 dodecahedral3 NON-CR YSTALLINE Ordered : icosahedron4 N = 13, 55, 147, 309,561... pentagonal bipyramid (this work) Random :

quenched liquid clusters relaxed hard-sphere models (this work)

ter) the stable configurations under a reasonable potential — although it is not in general possible to prove that all such configurations have in fact been found. Such studies have been carried out by Mclnnes [5, 6] and Bonissent and Mutaftschiev [7] for the Lennard-Jones (12-6) and Morse (a = 3) potentials. The number of stable isomers is found to increase steeply with the number of atoms and becomes unreasonable to calculate for more than

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C2-38 J. A. BARKER

about thirteen atoms. For larger assemblies one must be content with trying to identify the most important regi6ns of configuration space ; that is, clusters of high stability or which are very likely to form in a growth sequence.

One can obviously cut a small assembly of atoms out of a crystal lattice and thus form clusters of greatly varying surface geometries, for example the 561-atom f .c.c. cluster shown in figure 1. The types

I.I(,. 1. - A i O l - , r t o m cuboctahedral f.c.c. cluster relaxedunder a Lennard-Jones (12-6) potential.

listed in table I are the most obvious and therefore the most used ; they would also be expected to include the most stable crystalline forms though the geometry needed to achieve this will vary with size.'

Since the observations and descriptions of multiply-twinned particles in deposited metal clusters [8-1 11, two non-crystalline but highly symmetrical forms of cluster model have been considered - namely the icosahedron and pentagonal bipyramid. In figure 2 are depicted the smallest of the set of

FIG. 2. - The smallest of the icosahedral particles containing 13, 55, 147, 309 and 561 atoms. The largest is drawn from

relaxed coordinates.

icosahedral particles consisting of a central atom plus one to five shells.

The second category of non-crystalline models included in table I, the random models, is hard to define (in the same way as it is hard to define an amorphous solid). However, it is clear that one can obtain small assemblies of atoms which are neither crystalline nor multiply-twinned (as would be clear from observations of the interference functions, for example). Two candidates are proposed as serving as models of such clusters ; these are discussed later in the paper and a third type is introduced.

3. Comparison of relaxed icosahedral and cubocta- hedral structures.

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It is of some interest to compare the two 561-atom clusters shown in figures 1 and 2. Both clusters are drawn from relaxed coordinates using the Lennard-Jones (12-6) potential. By the term relaxed we mean that the atom positions are such that the potential energy of the system is at a minimum with respect to the atomic coordinates. These minima were computed using the Fletcher-Reeves conjugate gradients opti- mization procedure [l2] having assumed hard- sphere coordinates as a starting position in the 1677 (3 N-6) dimensional space.

We note the apparently undistorted planar faces of both relaxed clusters. The following are the -

radial distances between the apices of successive shells of the icosahedron, measured in units of the distance at which the L.J. potential is a minimum :

0.908, 0.928,0.946,0.959, 0.987. For the cuboctahedral cluster the corresponding distances are : 0.971, 0.971, 0.972, 0.976, 1.000. Here the compression at the centre of the cluster is far .less than for the icosahedron and in fact corresponds to the atomic nearest-neighbour distance for a bulk L.J. system, i.e. 0.9712. We also note the relative expansion of the outer shell of the cluster ; it should be realized, however, that the surface bonds of the cluster are in compression as a result of non-nearest-neighbour forces.

Using the well-depth of the L.J. potential as a unit of potential energy, it is found that the binding energy per atom of the 561-atom icosahedral structure is 6.85 compared with 6.76 for the cuboctahedral form

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although this is probably not the most stable f.c.c. cluster of this number of atoms.

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which are more stable than any in the previously described classes.

What criteria should we use in designing clusters ? First of all we must demand that they be locally dense and this in turn suggests that they should have a high tetrahedral content. Secondly, it seems reasonable to require that they should be stable under some realistic potential. As a rule the L.J. potential was use in this context ; not only is this potential particularly suitable in the description of rare-gas systems [I31 but, perhaps more impor- tantly, it is easily computed.

Our first criterion suggests that we should try to build large structures from units which are themselves known to be particularly stable and consequen- tly highly probably in a growth sequence. It is, therefore, not inappropriate to continue with a brief discussion of the early stages of cluster growth.

5. The early stages of cluster growth.

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Figure 3 is taken from a paper by Burton [14] and is based on the work of Hoare and Pal [IS, 161. It is intended to represent an atom-by-atom growth sequence for atoms under a simple pair potential. The sequence leading to the 13-atom icosahedron (Fig. 3h) is well favoured in relation to any other process - passing

FIG. 3 . -The early stages of cluster growth (after Burton [14]).

through the configurations likely to be of minimum energy at each size. The 13-atom f .c.c. structure (Fig. 39 and h.c.p. structure (Fig. 33) are both unlikely to form because this would require an atom to be placed in contact with just two others at some stage ; in any case these clusters are found to be unstable under either L.J. or Morse potentials, collapsing spontaneously to the icosahedral form. Although the growth of the 33-atom dodecahedra1 cluster (Fig. 3k) does seem quite probable, with

20 atoms being placed on the triangular faces of the 13-atom icosahedron, the subsequent stages of the sequence are far less certain. The 44-atom cluster (Fig. 31) is stable under a L.J. potential but is somewhat distorted. It would seem reasonable that, under most experimental conditions, after the addition of some ten atoms to the dodecahedra1 cluster (Fig. 3k), the resulting cluster would trans- form (by passing over some relatively low saddle- point) to the 43-atom structure shown in figure 4, which in fact has a lower potential energy (L.J.) per atom than the 45-atom cluster. This 43-atom icosidodecahedral cluster can be thought of as a 13-atom icosahedron with an octahedron on each face (described as a deltahedron by Cundy [ l q ) or,

alternatively, as a 55-atom icosahedron with its outermost twelve atoms removed.

FIG. 4. - The 43-atom icosidodecahedral cluster.

It has been thought [2] that the 55-atom cuboctahedron (Fig. 3n) is unstable under a L.J. potential

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C2-40 J. A. BARKER 6. Packings of icosahedra.

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Returning to the

problem of designing clusters, it is now clear that the 13-atom icosahedron is particularly suitable as a structural subunit.

Icosahedra can be fitted together, face-to-face (three possible angles for three icosahedra) and six icosahedra can thus be built into a ring with little distortion, or five icosahedra can be built into a tetrahedral unit without distortion. Having constructed such clusters, both as physical models and in the computer, we found them difficult to extend in any meaningful way. (One interesting discovery, however, was that if icosahedra are fitted together with an octahedron between each pair, then one can construct a Wurzite lattice of icosahedra).

Icosahedra are well-known in studies of interme- tallic compounds (IMC's)

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notably through the work of Samson [l8, 191, where a ring of five 13-atom icosahedra (point-to-point) is a frequently encountered feature. The addition of two further icosahedra to such a ring results in a 75-atom pentagonal bipyramid of icosahedra. We extended these ideas by adding six more icosahedral units in such a way as to produce a 127-atom icosahedron of icosahedra (Fig. 5)

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a structure which, to our knowledge, does not appear in the IMC literature.

V

FIG. 5. - The 127-atom icosahedron of icosahedra.

Removing the outermost 12 atoms from this assembly we obtain the 115-atom cluster shown in' figure 6a ; all the surface atoms of this cluster assume equivalent positions and the surface geometry is that of a rhombicosidodecahedron. Removal of these surface atoms will reveal an only slightly distorted 55-atom icosahedron.

This 115-atom cluster seems to be of particular importance. To begin with, its L.J. binding energy

per atom (5.66) is only slightly less than that of the most stable cluster (Fig. 6b) we have been able to find of about this size (5.71). This is a 116-atom cluster obtained by removing a cap of 31 atoms from the 147-atom icosahedron. Using a Morse

(a = 3) potential, the 115-atom cluster (binding energy = 9.72 well-depth units) is more stable than the 1 16-atom structure (9.53 units).

Further evidence for the importance of this 115-atom cluster is provided by the results of computer experiments. L e e , Barker a n d Abraham 1201 performed a Monte Carlo simulation using a L.J. potential and presented a picture of a 100-atom cluster near 0 K (Fig. 6c). Careful inspec- tion of the surface atoms and facets reveals that this is an incomplete form of the rhombicosidodeca- hedral cluster. A second example is shown in figure 6d which is a 129-atom cluster produced in a molecular dynamics simulation of the quenching of a liquid cluster performed by a group at the Facult6 des Sciences d'Orsay

.

It should be noted that these are not specially selected diagrams since investigators in these fields rarely publish drawings of particular configurations. 7. Extension of icosahedral packings.

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The geometrical fact that makes it possible to construct the 127-atom cluster, as described above, is that twelve slightly distorted dodecahedra can be packed around a regular dodecahedron. This construction forms the centre of what is known as the radial net [21]. We thus see that sphere packings

with dodecahedra1 symmetry can be built up to form larger units provided the spheres are soft enough to accommodate the necessary distortion. The 33-atom cluster shown in figure 3k is the most obvious to have the required symmetry, 13 such units were built (in the computer) into a 299-atom cluster. However, on relaxation this cluster underwent severe internal distortions so that the original geometry was changed considerably. It was found that the 43-atom cluster (Fig. 4) could also be built-up according to these principles. The resulting 399-atom structure (Fig. 7) is particularly stable and very little atomic movement is observed on relaxation under a L.J. potential. By filling the exposed pentagonal facets of the subunits of this cluster, we obtain the 471-atom .cluster shown in figure 8. This remarkable structure may be regarded as thirteen interpenetrating 55-atom icosahedra. Alternatively, one may choose to note that the nearly planar rhombic faces together constitute a surface geometry of a rhombic triacontahedron, as is emphasized in the figure.

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FIG. 7. - A 399-atom cluster built from thirteen of the 43-atom

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J. A. BARKER

w

FIG. 9. - The 279-atom cluster contained in the 471-atom cluster (figure 8).

apparently not drawn or constructed, by J.D. Bernal.

The 115-atom rhombicosidodecahedron (Fig. 6a) can also be used as a dodecahedra1 subunit, thirteen such units forming a 1285-atom structure ; the gaps between the units are trigonal prisms and tetrahe- dra. On relaxation under the L.J. potential, however, this very large structure undergoes considera- ble atomic rearrangement. A softer potential may not have caused this distortion.

8. Alternative use of the radial net.

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Here we further demonstrate that the radial net can provide a framework for the building of large clusters. The 26-atom cluster shown in figure 10a is built by adding atoms to a tetrahedral seed while maintaining the tetrahedral symmetry. It was men- tioned by Boerdijk [22] and can be thought of as four interpenetrating 13-atom icosahedra. Five such units form a ring (Fig. lob) and twenty form a

dodecahedron, the centre of which can be filled with a 55-atom icosahedron. We thus obtain an assembly of 485 atoms (Fig. 10c) which constitutes the central unit of a radial net which, if extended to the next shell, would consist of some four thousand atoms.

No attempt is being made to maintain that such configurations would be commonly found in monatomic systems. It does seem likely, however, that the geometric principles that make it possible for us to design these densely packed aggregates will be taken advantage of by nature.

9. Clusters in amorphous assemblies.

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The second part of this paper is concerned with the possibility of the appearance of non-crystalline clusters in amorphous materials. This will be discussed in the context of the work of Briant and Burton [23-251 whose thesis is that the 13-atom

FIG. 10. - (a) The 26-atom tetrahedral cluster.

( b ) A ring of five such clusters.

( c ) Dodecahedra1 485-atom structure built from twenty of the

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icosahedron may be the underlying structural unit of amorphous metals. This contention is supported by three types of evidence :

Firstly, on comparing the Debye-Scherrer interference functions (IF's) calculated for the 13-atom icosahedron and those measured for a number of amorphous materials ; Briant and Burton note the similarity with regard to the shoulder on the second peak. Secondly, they observed the 13-atom icosahedron in two clusters (or 19 and 55 atoms) which they had quenched from the liquid state in a molecular dynamics simulation. These two clusters had IF's with the characteristic second peak.

The third piece of evidence is the simple but important expectation that the atom-by-atom growth sequence should lead to the 13-atom icosahedron.

A natural extension of their work would be computer experiments on bigger clusters to simu- late the formation of amorphous aggregates under various conditions. One such experiment was described very recently by Rahman et a1. 1261, although no I F was presented. Doubtless this will lead to much more work, particularly because of the significance of such experiments in understanding the glass-transition. In the meantime we can content ourselves with the study of other amorphous models

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namely the dense random packed structures.

10. Random packings of soft spheres.

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Probably the best known dense random packing model is that described by Finney [ 2 7 which is a ball-bearing model' of the type introduced by Bernal [28]. After the construction of this model, the positions of the 7994 ball-bearings were carefully measured so that the coordinates could be analysed by computer.

In collaboration with Finney, we investigated the effect of relaxing the central 999 atoms of this model under a L.J. potential [29]. Although one might hope that the resulting cluster would be typical of amorphous clusters of about the same size, there are two points that should be noted. Firstly, it was evident from drawings of the cluster that there existed a certain amount of structural anisotropy both before and after relaxation (this feature has recently been discussed by Alben et al. [30]). Secondly, since we saw little structural rearrangement on relaxation, the relaxed cluster might still be more typical of a hard rather than soft-sphere packing, containing what might be termed jammed configurations.

With these problems in mind, we built, for comparison, a random soft-sphere model using the computer. Starting with a tetrahedral seed cluster, atoms were deposited isotropically onto the cluster which was then relaxed under the L.J. potential. The relaxation was performed after the addition of each atom until the cluster contained 10 atoms and

subsequently after sufficient atoms had been deposited to cover about one quarter of the existing cluster. A 509-atom assembly was thus obtained, and, as had been hoped, drawings did not reveal any anisotropy

.

In figure 11 are shown pair distribution functions (PDF's) for the centre of the Finney model, both before and after relaxation, and for the new random soft-sphere model\. Relaxation of the hard-sphere model results in a splitting of the second peak of the PDF. This feature, which is also apparent for the 509-atom cluster, is a well-known characteristic of amorphous materials [31, 321. Computer-built hard-sphere models [33, 341 do not exhibit a split second peak.

r/ro

FIG. 11.

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Pair distribution functions for the 999-atom core of the Finney model, before (top) and after (middle) relaxation, and

for the 509-atom soft-sphere model (bottom).

11. Interference functions.

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We are now in a position to present and discuss the Debye-Scherrer interference functions (IF's) for the various model assemblies described above. These functions were calculated according to the formula [35]

I ( k ) = 1

+2

[

sin 2 vkrO cj

N

2 vkro rij

I

where

N

is the number of atoms and i-ti is the

distance between atoms i and j measured in units of the position of the minimum of the L.J. potential, ro.

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C2-44 J. A. BARKER

FIG. 12. - Debye-Scherrer interference functions calculated for

( a ) the 999-atom core of the Finney model, ( b ) the relaxed

Finney model and ( c ) the 509-atom random soft-sphere model.

In order to investigate the changes in internal geometry that had occurred during the relaxation of the Finney model, the statistics of the Voronoi polyhedra [27] were computed [26]. This revealed that the number of 13-atom icosahedra within the structure had approximately doubled.

Random hard-sphere models were built in the computer by Sadoc et al. [36]. These models were unusual in that they included a proportion of small spheres with the restriction that no two such spheres could be neighbours. This algorithm favours the appearance of the 13-atom icosahedron and the calculated Debye-Scherrer functions had distinct shoulders on the second peak.

I I

0 1 2 3 4 5

k re

FIG. 13. - Debye-Scherrer interference functions for some of the smaller clusters shown in ( a ) figure 10a, ( b ) figure 4, ( c )

figure 3m, and (d) figure 6a.

In figures 13 and 14 are shown IF'S for some of the ordered non-crystalline clusters described earlier. As was observed by Briant [25], the shoulder is not apparent in the I F for the 55-atom icosahedron and the same is true for the other multiply-twinned cluster of 561 atoms. The 115- atom rhombicosidodecahedron (Fig. 6a) does show this feature in its I F which is, of course, very similar to that of the 129-atom quenched cluster (Fig. 6d).

1

0 1 2 3 4 5

k ro

FIG. 14.

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Debye-Scherrer interference functions for the clusters shown in ( a ) figure 8, ( b ) figure 2 , and ( c ) figure 1 .

Perhaps the most interesting of these patterns is that for the 471-atom triacontahedron (Fig. 8)

which is extremely similar to those of the relaxed random structures. We know, from its construction, that this cluster contains thirteen 13-atom icosahedra. (Its PDF, not shown here, is also similar to those of the relaxed random structures.)

12. Discussion.

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It is clear that the above observations add weight to the proposition that the 13-atom icosahedron is an important structural unit in amorphous solids. The appearance of this unit is encouraged by the application of a realistic potential to models and, with its increase in proportion, both the PDF and I F become more similar to those of amorphous materials.

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such ordered regions could well provide a geometric explanation of the glass transition.

No apologies are made for having tried to draw conclusions about the extended phase from studies of finite systems, since the introduction of periodic boundary conditions would seriously impair the appearance of the 5-fold symmetry so important in our considerations.

13. Summary.

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In the first part of this paper a brief description was given of the kinds of models of small monoatomic clusters that have commonly been used as a basis for calculations and for comparison with experimental observations. It was demonstrated that, by using this set alone, certain important regions of configuration space would be neglected ; models representing these regions were designed, using soft rather than hard spheres, within the geometrical framework of the 56 radial net.

The second part of the paper was concerned with the appearance of non-crystalline clusters in amorphous materials. This provided an opportunity to introduce models of random spherical clusters. The properties and internal geometries of these structures were investigated and compared with those of the designed clusters. Several of these were found to exhibit very similar properties and it was concluded that there could exist large ordered regions in amorphous materials.

Acknowledgments. The author would like to thank the organizers of the International Meeting on Small Particles and Inorganic Clusters for inviting him to present this paper.

Much credit is due to Dr. M. R. Hoare who inspired much of the work described here.

This work was supported by a grant from the Science Research Council.

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