lia

lly isresented.

n whichhe Letter

Physics Letters A 319 (2003) 373–378

www.elsevier.com/locate/pla

Interlocking properties of buckyballs

A.V. Dyskina,∗, Y. Estrinb, A.J. Kanel-Belovc, E. Pasternaka,b

a School of Civil and Resource Engineering, The University of Western Australia, 35 Stirling Highway, Crawley WA 6009, Austrab Institut für Werkstoffkunde und Werkstofftechnik, Technische Universität Clausthal, D-38678 Clausthal-Zellerfeld, Germany

c School of Engineering and Science, International University of Bremen, P.O. Box 750 561, 28725 Bremen, Germany

Received 16 September 2003; accepted 6 October 2003

Communicated by V.M. Agranovich

Abstract

A geometric possibility of arranging C60 molecules (buckyballs) in a monolayer in which they are interlocked topologicaconsidered. Three arrangements of this type, all different from the common, face centred cubic fullerene structure, are pAlthough in this analysis the buckyballs are treated in a simplified way, as rigid bodies, the realization of structures iC60 molecules would be arranged in the way described may lead to mechanically stronger fullerene monolayers. Taims to initiate discussion of the viability and the potential of the new systems of interlocked buckyballs. 2003 Elsevier B.V. All rights reserved.

PACS:61.48.+c; 81.05.Tp

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1. Introduction

Geometry of C60 molecules, in particular their symmetries and associated properties, have been esively studied [1]. Geometrically, this molecule isbuckyball—a truncated icosahedron with symmeaxes of fifth, third and second orders. In a recpublication [2] we suggested that the buckyballs,ing truncated icosahedra, inherit from the platobodies an interesting geometric property unknowntil then: identical elements shaped as one of the

* Corresponding author.E-mail address:adyskin@cyllene.uwa.edu.au (A.V. Dyskin)

0375-9601/$ – see front matter 2003 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2003.10.027

-

platonic bodies can be assembled in a layer-like stture in which they are interlocked.

In this Letter we demonstrate the existencethree interlocking arrangements of buckyballs. Thform a set of three layer-like systems which athe only possible, qualitatively different interlockinarrangements of buckyballs.

Interlocking is a property of solid bodies. Consquently, geometrical and mechanical properties ofterlocking structures have so far been studied for sbodies [2–4]. It is still an open question whether siilar assemblies of molecules are energetically admsible and if so, what properties such assemblies mpossess. The aim of this Letter is to draw the attenof the physics community to the geometric possibiof existence of assemblies of interlocked buckyb

.

374 A.V. Dyskin et al. / Physics Letters A 319 (2003) 373–378

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and to initiate discussion of potential ways to utilithis possibility.

2. Interlocking of convex polyhedra in a layer

Historically, the first interlocking structure founindependently by Glickman [5] and then by the presauthors [3] was a layer-like assembly of tetrahedThis assembly, Fig. 1(a), being the simplest oneconvenient to use for explaining the principletopological interlocking. This assembly may haveimportance of its own, since some molecules htetrahedral structure.

Each tetrahedron in the assembly under consiation has one of its second order symmetry axesented normal to the layer. The tetrahedron cont

(a)

(b)

Fig. 1. Principle of interlocking of tetrahedral elements demstrated by (a) a fragment of interlocking assembly, (b) centraltion of the assembly with arrows indicating the direction of tralation of the section sides when the section moves upwards.neighbouring elements numbered 1 and 3 block the upwardplacements of the reference element (R), while elements 2 aobstruct its downward displacements.

four neighbours. For a reference tetrahedron R shin Fig. 1(a), they are numbered 1–4. Neighbourand 3 block the reference tetrahedron from beingplaced upwards, while neighbours 2 and 4 prevendownward displacements. Fig. 1(b) shows the midsection of the assembly. The arrows indicate the ditions in which the sides of the sections of the refence tetrahedron (and those of the neighbours intact with it) will move when the section plane is shiftupwards, away from the middle section. In essenthese arrows show the directions of inclination offaces of the adjacent tetrahedra. The inclinationssociated with the inward directed arrows block theward displacements of the reference tetrahedron, wthe inclinations associated with outward directedrows block its downward displacements. Therefosuch alternating directions of the arrows (alternatinclinations of the faces) ensure interlocking.

We will also use the following criterion for translational interlocking. Consider polygons producedthe intersections of the extensions of the element faconstrained by the element’s neighbours with a plparallel to the layer. It is proven in Appendix A than element is (translationally) locked if, and onlyby continuously shifting the section plane in eitherrection normal to the middle section, the correspoing polygon eventually degenerates to a segmentstraight line or to a point.

The above-mentioned principles will now be usto explain the interlocking arrangements of bucballs.

3. Interlocking of buckyballs

3.1. Interlocking normal to symmetry axis of fifthorder

Consider the assembly of buckyballs in a layer ppendicular to afifth order symmetry axis. A middlesection normal to this axis is a decagon with the slength 3a/2, wherea is the edge length of a buckyball. An arrangement of decagons on a plane thatsures interlocking of buckyballs is shown in Fig. 2(Traces of the faces of the reference buckyball in ctact with its neighbours and the corresponding arroare marked by lettersc, d, . . . , h placed outside thebuckyball section. At first glance it appears that

A.V. Dyskin et al. / Physics Letters A 319 (2003) 373–378 375

theddles ofardsent

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(a)

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Fig. 2. Interlocking arrangement of buckyballs associated withfifth order symmetry axes: (a) arrangement of the decagonal misections of the buckyballs with the arrows showing the directionmovement of the decagon’s sides when the section moves upw(indicative of the inclinations of the contacting faces), (b) a fragmof interlocking assembly. The buckyballs contact each other onhexagonal faces.

reference buckyball can be removed by displacemfor example, in the direction of arrowsc andf . This is,however, not the case, since, as shown in Appendiby progressive shifting of the section plane, the cosponding polygon will eventually degenerate.

In the arrangement considered the buckyballssituated at the nodes of a deformed hexagonal la

and contact their neighbours on their hexagonal faThe deformed hexagonal lattice is characterizedtwo distances,d(1)5,6 and d(2)5,6, between the decagocentres as well as by the anglesϕ andψ , Fig. 2(a).These parameters are

d(1)5,6 = 3a

2

√5+ 2

√5, d

(2)5,6 = 3a

4

√26+ 6

√5,

(1)ϕ = cos−1

(d(1)5,6

2d(2)5,6

), ψ = π − 2ϕ.

Here the first index ind refers to the symmetry axithe layer is normal to and the second one refers tocontacting faces, the hexagons in this case.

A fragment of the assembly of this type is shownFig. 2(b).

3.2. Interlocking normal to symmetry axis of thirdorder

Consider now the assemblies of buckyballs inlayer perpendicular to athird order symmetry axis.A middle section normal to a third order axis is18-gon in which the sides are grouped in six tripleall six together form a hexagon-like figure, as seenFigs. 3(a) and 4(a). The middle side in each triphas the length ofa(

√5 + 1)/4. It passes through

pentagonal face of the buckyball. Two other sidesa triplet have the length ofa

√3/2 and pass throug

hexagonal faces of the buckyball. Figs. 3(a) and 4show two possible arrangements of the 18-gonsa plane that ensure interlocking. The arrangemof alternating inward and outward directed arrowsthese two cases make the translational interlockobvious.

In the first arrangement the buckyballs conteach other with their pentagonal faces. The distabetween the centres of the neighbouring 18-gand, correspondingly, the centres of the neighboubuckyballs is

(2)d3,5 = a

2

√52+ 18

√5.

In the second arrangement the buckyballs coneach other with their hexagonal faces. The distabetween the centres of the neighbouring buckyball

(3)d3,6 = a√

3(17+ 6

√5).

2

376 A.V. Dyskin et al. / Physics Letters A 319 (2003) 373–378

thethe

gonatheoves(b)

wayd.

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thethe

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allll asies

ico-ing

(a)

(b)

Fig. 3. Interlocking arrangement of buckyballs associated withthird order symmetry axes and characterized by contact onpentagonal faces: (a) the corresponding arrangement of the 18-middle sections of the buckyballs with the arrows showingdirections of movement of the 18-gon sides when the section mupwards (indicative of the inclinations of the contacting faces),a fragment of interlocking assembly. In order to make clear thethe buckyballs are interlocked, they are shown slightly separate

In both cases the buckyballs are arranged in hexonal lattices with different lattice sizes. In the secotype of lattice, the buckyballs are rotated clockwby 19.78◦. Fragments of the corresponding assembare shown in Figs. 3(b) and 4(b).

l

(a)

(b)

Fig. 4. Interlocking arrangement of buckyballs associated withthird order symmetry axes and characterized by contact onhexagonal faces: (a) the corresponding arrangement of the 18-middle sections of the buckyballs with the arrows showingdirections of movement of the 18-gon sides when the section mupwards (indicative of the inclinations of the contacting faces),a fragment of interlocking assembly.

4. Discussion and conclusion

Interlocking discussed above is strictly speaktranslational, since it was shown that no buckybcan be removed from the assembly by translationsprinciple, a combination of rotations and translatiocould be applied. It is shown in Appendix C that foraforementioned arrangements of buckyballs as weinterlocking arrangements for five platonic bod(tetrahedron, cube, octahedron, dodecahedron andsahedron) discussed in [2] translational interlock

A.V. Dyskin et al. / Physics Letters A 319 (2003) 373–378 377

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is tantamount to full interlocking, i.e., interlockinwith respect to any combination of translations arotations.

Thus, the geometric possibility of three typesarrangement of buckyballs into interlocking layer-listructures has been demonstrated. These arrangeare associated with the symmetries of fifth and thorder. Despite the presence of symmetry axessecond order, no interlocking arrangements associwith this type of symmetry are possible.

When several layers are put together, one onof another, some faces of the buckyballs belonginneighbouring layers may be in contact preventing mtual sliding of the layers. However, there is no intlocking in planes other than the planes of the laythemselves. In this sense, no three-dimensional sture with full interlocking can be formed with C60molecules. Still, the principal possibility of occurrenof monomolecular fullerene layersof the types de-scribed is an issue to be explored further.

Interlocking is a pure geometrical property thgoverns the interaction betweensolid bodies. Whilethe C60 molecules are not solid bodies, geometricasimilar arrangements of these molecules, if attainamight possess some unexpected physical propenotably higher mechanical strength, which warrafurther investigation of such structures. In particuit would be of interest to study the viability of sucstructures vis-à-vis that of the known, face centcubic, modification by comparing their energetics.

Acknowledgements

Support from the Australian Research Counthrough Discovery Grant DP0210574 (2002–20and Linkage International Grant LX0347195 (2002004) as well as DFG Grant ES-74/10-1 is acknoedged. One of the authors (E.P.) acknowledgesport through an Alexander von Humboldt ReseaFellowship 2002–2003.

Appendix A. Criterion for translationalinterlocking

Consider a reference polyhedron surroundedits neighbours. Continuation of each contacting fa

ts

,

of the polyhedron to a plane forms a boundarythe half space of permitted directions of translatioTherefore, the contacting faces of the polyhedronparallel to what can be called the cone of permitdirections of movement. This cone is not empty if, aonly if, the continuations of contacting faces of tpolyhedron form an unbounded region. Furthermothe boundness of this region is equivalent tofollowing. A polygonal section of the polyhedroformed by a plane parallel to the middle plane ofassembly degenerates to a straight segment or aas the secant plane shifts away from the middle pla

Appendix B. Proof of interlocking of buckyballsin Fig. 2

Suppose the section in Fig. 2(a) moves withconstant velocity in the direction normal to the plaof drawing such that the sides of the polygon insection move in the directions indicated by the arroDue to the buckyball symmetry the polygon sides wmove in these directions with a constant velocityv.Consider the common point of intersection of tcontinuations of sidese and g with the broken lineperpendicular to sidec. This point will move in thedirection of arrowc with velocity v/sinϕe, whereϕe is the angle between the broken line and sideeand g. This velocity is obviously greater thanv, sothat sidese andg will eventually catch up with sidecand the polygon will degenerate to a point. Therefoaccording to Appendix A, this buckyball structuretranslationally interlocked.

Appendix C. Condition of equivalence oftranslational interlocking to full interlocking

Full interlocking of the assembly shown in Fig.can be deduced from the following general statemFor assemblies of identical polyhedra translatiointerlocking is tantamount to interlocking in gene(meaning that an element cannot be removed bycombination of translations and rotations) if

(1) all faces that block the movement in one directare equally inclined to the middle plane, and

378 A.V. Dyskin et al. / Physics Letters A 319 (2003) 373–378

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(2) one can inscribe a circle in the middle sectionthe polyhedron such that the circle touches poof the contacting sites. (All interlocking arrangments of platonic solids except for icosahedas well as the buckyball arrangements shownFig. 3 satisfy this property.)

Based on the obviously valid statement that an elemis locked in a structure if a part of it is locked, wconsider a thin layer cut out from the assembly by tparallel planes below and above the middle section

We now prove that a reference polyhedronP cutout from the element in question by the above sectiis locked. Consider the faces ofP through which theneighbouring elements prevent its upward movemPut a sphere inP in such a way that it touchethese faces at points which are in contact withneighbouring elements. (This can be done as folloConsider the circle inscribed in the polygon in tmiddle section. Now consider a sphere with this cirbeing its great circle and shrink and lift it a bitinscribe it in the polyhedron in the thin layer.is always possible for thin enough layer, sinceoriginal circle touches the neighboring polyhedrainner points.) Similarly, put another sphere touchthe faces that preventP from moving downwardsThese two spheres do not coincide. No movemenP can displace the centre of the first sphere downthat of the second sphere up, otherwise the spabetween them would change (which is prohibitsince rigid body motion preserves distances.) Tmovements of the sphere centres consistent with rbody motion are precisely those movements thatblocked by the neighbouring elements. ThereforePis locked, and so is the element containingP , whichcompletes the proof.

Consider now the assembly presented by FigThe general statement cannot be used here beccondition (1) is violated since, for example, amonfacesc, e, andg that ensure the upward interlockinfacese and g form a sharp angle with the middplane, while facec forms an obtuse angle. Firstly wnote that hereafter we only have to consider rotatiabout axes parallel to the middle cross-section sall rotations in this plane are obviously blocked. Thwe note that for the assembly in Fig. 2 a rotation

e

the reference buckyball about an axisR normal toline cf is blocked. Indeed, the plane contact areafacesc andf prevent the rotation, since each of thecontains the base of normal drawn from the cenof the reference buckyball. Now suppose that this an axis about which rotation is admissible. Thbecause the arrangement possesses a plane of msymmetry passing throughcf, there must be anotheaxis of admissible rotation oriented symmetrically.we perform two successive infinitesimal rotations,ω1andω2, about these axes, we will obtain an admissirotation,ω1 + ω2. However, due to the symmetry thsum is an infinitesimal rotation aboutR, which wasshown to be blocked. Therefore, the buckyballs shoin Fig. 2 are rotationally and hence fully interlockeSince a buckyball is a truncated icosahedron,proof is applicable to the arrangements of interlocicosahedra as well.

For the assembly shown on Fig. 4 the general stment does not work since condition (2) is violated. Wnote, however, that rotations about an axisR parallelto the faces contacting neighbours 1 and 4 are blocby other neighbours. Suppose there is an axis awhich a rotation is admissible. Because of the symetry of 3rd order, there must exist another axisadmissible rotation, at an angle of 2π/3. These admissible infinitesimal rotations form a basis, hence thexists a linear combination of them, also admissibwhich is a rotation aboutR. Since rotations aboutRwere shown to be blocked, there are no admissibletations and the assembly is rotationally interlocked

References

[1] M.S. Dresselhaus, G. Dresselhaus, P.C. Ecklund, SciencFullerenes and Carbon Nanotubes, Academic Press, New Y1995.

[2] A.V. Dyskin, Y. Estrin, A.J. Kanel-Belov, E. Pasternak, PhiloMag. Lett. 83 (2003) 197.

[3] A.V. Dyskin, Y. Estrin, A.J. Kanel-Belov, E. Pasternak, ScripMater. 44 (2001) 2689.

[4] A.V. Dyskin, Y. Estrin, A.J. Kanel-Belov, E. Pasternak, AdEng. Mater. 3 (2001) 885.

[5] M. Glickman, in: Proceedings of the Second InternatioConference on Concrete Block Paving, Delft, April 10–11984, p. 345.