11

Fullerenes 1Fullerenes 1

MirceaMircea V. DiudeaV. Diudea

Faculty of Chemistry and Chemical EngineeringFaculty of Chemistry and Chemical EngineeringBabesBabes--BolyaiBolyai UniversityUniversity400084400084 ClujCluj, ROMANIA, ROMANIA

diudea@chem.ubbcluj.rodiudea@chem.ubbcluj.ro

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ContentsContents

•• Fullerenes Fullerenes –– ShortShort HistoryHistory

•• Basic Relations in PolyhedraBasic Relations in Polyhedra

33

Fullerenes: Short HistoryFullerenes: Short History(by (by KrotoKroto11))

1. Kroto1. Kroto, H. The first predictions in the Buckminsterfullerene crystal b, H. The first predictions in the Buckminsterfullerene crystal ball. all. Fuller. Fuller. SciSci. . TechnolTechnol. 1994, . 1994, 22, 333, 333--342.342.

44

Before HistoryBefore History

•• The classic text of D’Arcy Thompson “The classic text of D’Arcy Thompson “On On Growth and FormGrowth and Form”,”, Cambridge Univ. Press Cambridge Univ. Press (1942), speaks about an (1942), speaks about an AuloniaAulonia hexagonahexagona,,a sea creature with a a sea creature with a silicioussilicious skeleton.skeleton.

55

AuloniaAulonia hexagonahexagona (by (by HaeckelHaeckel))

66

Before HistoryBefore History

•• Jones, D. E. H. Jones, D. E. H. NewNew ScientistScientist, 1966, 32, p. 245.; , 1966, 32, p. 245.;

•• JonesJones, D. E. , D. E. H., H., TheThe Inventions of Inventions of DaedalusDaedalus,,

Freeman: Oxford, 1982, pp. 118Freeman: Oxford, 1982, pp. 118--119.119.

““the high temperature graphite production might the high temperature graphite production might be modified to generatebe modified to generate graphitegraphite balloonsballoons”.”.

77

First First ConsiderationsConsiderations

•• EijiEiji OsawaOsawa,, KagakuKagaku (Kyoto) 1970, (Kyoto) 1970, 2525, 854, 854--863 863 ((in in JapaneseJapanese););ChemChem. . AbstrAbstr. 1971, . 1971, 7474, , 75698v.75698v.

(The original conjecture of a stable C(The original conjecture of a stable C6060 molecule). molecule).

Yoshida, Z.; Yoshida, Z.; OsawaOsawa, E., E. AromaticityAromaticity. . KagakudojinKagakudojin: Kyoto, : Kyoto, 1971 (in Japanese)1971 (in Japanese)

BochvarBochvar, D. A.; , D. A.; Gal’pernGal’pern, E. G. , E. G. DoklDokl. . AkadAkad. . NaukNauk SSSR,SSSR, 1973, 1973, 209209, 610, 610--612. (English translation, Proc. Acad. 612. (English translation, Proc. Acad. SciSci. USSR, . USSR, 1973, 1973, 209209, 239, 239--241).241).

(Huckel(Huckel calculations)calculations)

88

Eiji Osawa and Mircea DiudeaEiji Osawa and Mircea DiudeaOkazaki, Japan, Jan. 8, 2004Okazaki, Japan, Jan. 8, 2004

99

Calculations and NomenclatureCalculations and Nomenclature

•• Davidson, R. A.Davidson, R. A. TheorTheor. . ChimChim. . ActaActa, 1981, , 1981, 5858, , 193193--195.195.((HuckelHuckel calculations)calculations)

•• CastellsCastells, J.; , J.; SerratosaSerratosa, F. , F. J.,J., Chem. Chem. Ed.,Ed., 1983, 60, 1983, 60, 941. (ibid. 1986, 63, 630)941. (ibid. 1986, 63, 630)

(C(C6060 and Cand C6060HH6060 (IUPAC Nomenclature)(IUPAC Nomenclature)

•• HaymetHaymet, A. D. , A. D. J.,J., Chem. Phys Chem. Phys LettLett.,., 1985, 1985, 122122, 421, 421--424. 424. ((HuckelHuckel calculations)calculations)

1010

CC6060 –– First SynthesesFirst Syntheses•• KrotoKroto, H.; Heath, J. R.; O’Brian, S. C.; Curl, R. F.; , H.; Heath, J. R.; O’Brian, S. C.; Curl, R. F.;

Smalley, R. E.Smalley, R. E. (Nobel Prize(Nobel Prize--1995)1995)Sussex University (UK) & Rice University (Sussex University (UK) & Rice University (USA),USA),

Buckminsterfullerene Buckminsterfullerene CC6060 isolated from isolated from selfself--assemblyassembly

products of graphite heated by plasma.products of graphite heated by plasma.

NatureNature (London) , 1985, (London) , 1985, 318318, 162, 162--163.163.

•• KraetschmerKraetschmer, W.; Lamb, L. D.; , W.; Lamb, L. D.; FostiropoulosFostiropoulos, K.; , K.; Huffman, D. R.,Huffman, D. R., Solid CSolid C6060: a new form of carbon. C: a new form of carbon. C6060isolated in macroscopic amount by arc vaporization of isolated in macroscopic amount by arc vaporization of graphite.graphite.

NatureNature (London) , 1990, (London) , 1990, 347347, 354, 354--358.358.

1111

NanotubesNanotubes and and ToriTori

•• S. IijimaS. Iijima,, HelicalHelical microtubules of graphitic microtubules of graphitic carbon. carbon. NatureNature (London), 1991, (London), 1991, 354354, 56, 56--58.58.

•• Liu, J.; Dai, H.; Liu, J.; Dai, H.; HafnerHafner, J. H.; Colbert, D. T.; , J. H.; Colbert, D. T.; Smalley, R. E.; Tans, S. J.; Smalley, R. E.; Tans, S. J.; DekkerDekker, , C.,C., Fullerene Fullerene "crop circles". "crop circles". NatureNature,, 1997, 1997, 385385, 780, 780--781781

•• R. Martel, H. R. R. Martel, H. R. SheaShea, and Ph. , and Ph. AvourisAvouris,, Ring Ring formation in singleformation in single--wall carbon wall carbon nanotubesnanotubes..

J. Phys. ChemJ. Phys. Chem. . B,B, 19991999, , 103103,, 75517551--7556.7556.

1212

Mircea Diudea and Sumio Iijima,Mircea Diudea and Sumio Iijima,Okazaki, Japan, Jan. 8, 2004Okazaki, Japan, Jan. 8, 2004

1313

Isolated FullerenesIsolated Fullerenes

•• N N = 20, 36, 60, 70, 76, 78, 82, 84, 96= 20, 36, 60, 70, 76, 78, 82, 84, 96

•• H. Prinzbach et al.,H. Prinzbach et al., Nature, 2000, 407, 60Nature, 2000, 407, 60--63 /63 / M. SaitoM. Saito and and

MiyamotoMiyamoto, , Phys. Rev. LettPhys. Rev. Lett., 2001, 87, 035503 /., 2001, 87, 035503 / J. Lu et al.,J. Lu et al., Phys. Phys. Rev. BRev. B, 2003, 67, 125415., 2003, 67, 125415.

•• C. Piskoti, J. Yarger, and A. Zettl,C. Piskoti, J. Yarger, and A. Zettl, NatureNature, 1998, 393, 771, 1998, 393, 771--773. 773.

•• R. Ettl, I. Chao, F. Diederich, R. L. Whetten,R. Ettl, I. Chao, F. Diederich, R. L. Whetten, NatureNature, 1991, , 1991, 353353, ,

149.149.•• F. Diederich, R. L. Whetten, C. Thilgen, R. Ettl, I. Chao, and F. Diederich, R. L. Whetten, C. Thilgen, R. Ettl, I. Chao, and M. M.

M. Alvarez,M. Alvarez, ScienceScience, 1991, , 1991, 254254, 1768., 1768.•• K. Kikuchi, N. Nakahara, T. Wakabayashi, S. Suzuki, H. K. Kikuchi, N. Nakahara, T. Wakabayashi, S. Suzuki, H.

Shiromaru, Y. Miyake, K. Saito, I. Ikemoto, M. Kainosho, and Y. Shiromaru, Y. Miyake, K. Saito, I. Ikemoto, M. Kainosho, and Y. Achiba,Achiba, NatureNature, 1992, , 1992, 357357, 142., 142.

1414

CC60 60 (N=12k; k = 5) (top)(N=12k; k = 5) (top)CC6060 ((IIhh)) (side)(side)

Fullerene Fullerene = Cage tiled with= Cage tiled with pentagonspentagons (12)(12)andand hexagonshexagons (N/2(N/2--10)10)

18121812 topological isomerstopological isomers

1515

CC60 60 NomenclatureNomenclature1,21,2

•• Hentriacontacyclo[29.29.0.0Hentriacontacyclo[29.29.0.02,142,14.0.03,123,12.0.04,594,59.0.05,105,10.0.06,56,5

88.0.07,557,55.0.08,538,53.0.09,219,21.0.011,2011,20.0.013,1813,18.0.015,3015,30.0.016,2816,28.0.017,2517,25.0.019,2419,24. .

0022,5222,52.0.023,5023,50.0.026,4926,49.0.027,4727,47.0.029,4529,45.0.032,4432,44.0.033,6033,60..

0034,5734,57.0.035,4335,43.0.036,5636,56.0.037,4137,41.0.038,5438,54.0.039,5139,51.0.040,4840,48..0042,4642,46] hexacontane.] hexacontane.

1. J. Castels, Some comments on fullerene terminology, nomenclature,and aromaticity. Fullerene Sci. Technol. 1994, 2, 367-379.

2. J. Castels and F. Serratosa, J. Chem. Ed., 1986, 63, 630.

1616

Basic Relations Basic Relations in Polyhedrain Polyhedra

•• First theorems on graph counting (First theorems on graph counting (EulerEuler))1,21,2

∑∑dd ( ( dvdvdd )) = 2= 2ee (1)(1)

∑∑ss ( ( sfsfss )) = 2= 2ee (2)(2)

where where vvdd andand ffss denote vertices of degreedenote vertices of degree d d andand ss--sized faces,sized faces,respectively.respectively.

1. Euler, L. Solutio Problematis ad Geometriam Situs Pertinentis. Euler, L. Solutio Problematis ad Geometriam Situs Pertinentis. Comment. Acad. Sci. I. PetropolitanaeComment. Acad. Sci. I. Petropolitanae 1736, 1736, 88, 128, 128--140.140.

2. King2. King, R. , R. B.,B., Applications of Graph Theory and Topology in Applications of Graph Theory and Topology in Inorganic Inorganic ClusterCluster and Coordination Chemistryand Coordination Chemistry, , CRCCRC Press, 1993.Press, 1993.

1717

f1

f2f3

f4

Planar GraphPlanar GraphA graph isA graph is planarplanar if it can be drawn in the plane without crossings.if it can be drawn in the plane without crossings.

Its regions are called Its regions are called facesfaces, , ff. The unbounded region is called. The unbounded region is calledthe the exteriorexterior face (face (HararyHarary).).11

A graph is planar if and only if it has no subgraphs homeomorphiA graph is planar if and only if it has no subgraphs homeomorphic c to to KK55 or or KK3,33,3 (Kuratowski).(Kuratowski).22

1. 1. Harary , F. Harary , F. Graph TheoryGraph Theory, Addison , Addison -- Wesley, Reading, M.A., 1969.Wesley, Reading, M.A., 1969.2. Kuratowski, K. Sur la Problème des Courbes Gauches en Topolog2. Kuratowski, K. Sur la Problème des Courbes Gauches en Topologie, ie,

Fund. MathFund. Math. 1930, . 1930, 1515, 271, 271--283.283.

1818

EulerEuler Theorem on PolyhedraTheorem on Polyhedra11

v v –– e + fe + f = = χχ = = 22((11 –– gg)) (3)

χχ = Euler= Euler’’s s characteristiccharacteristicv = v = number of vertices, number of vertices, e = e = number of edges,number of edges,f = f = number of faces,number of faces,gg = genus ; = genus ; ((gg = 0 for a sphere; 1 for a = 0 for a sphere; 1 for a torustorus))..

A A consequenceconsequence::A sphereA sphere can not be tessellated only by hexagons.can not be tessellated only by hexagons.

Fullerenes need Fullerenes need 12 pentagons12 pentagons (for(for closingclosing the the cage) and cage) and (N/2(N/2--10) hexagons10) hexagons..In the opposite, a tube In the opposite, a tube andand a a torustorus allow pure hexagonal nets.allow pure hexagonal nets.

1. L. Euler, Elementa doctrinae solidorum, Novi Comment. Acad. Sci. I. PetropolitanaeComment. Acad. Sci. I. Petropolitanae1758, 4, 109-140.

1919

Fullerene countingFullerene countingRewrite relations (1), (2), and (3) as:Rewrite relations (1), (2), and (3) as:•• 33vv = 2= 2ee (1’) (1’) •• 55ff55 + 6+ 6ff6 6 = 2= 2ee (2’)(2’)•• v v + + ff = 2 + = 2 + ee (3’)(3’)

•• Substituting v in (3’) by its value from (1’) one can write:Substituting v in (3’) by its value from (1’) one can write:

•• (2/3)(2/3)ee + + f f = 2 + = 2 + ee•• 22ee + 3+ 3ff = 6 + 3= 6 + 3ee•• ee = 3= 3f f –– 66 (4)(4)

•• Expressing Expressing ff by its composition: (by its composition: (ff = = ff55 + f+ f66), relation ), relation (4)(4) becomes:becomes:•• ee = 3(= 3(ff55 + + ff66) ) –– 66 (5)(5)•• Substituting e from (5) in (2’) one obtains:Substituting e from (5) in (2’) one obtains:•• 55ff55 + 6+ 6ff66 = 6(= 6(ff55 + + ff66) ) –– 1212

•• ff55 = 12= 12 (6)(6)

•• From (1’), (2’), and (6) the expression for From (1’), (2’), and (6) the expression for ff66 is obtined:is obtined:

•• 55ff55 + 6+ 6ff66 = 3= 3vv

•• ff66 = = vv/2 /2 –– 1010 (7)(7)

2020

By substituting By substituting vv, , ee and and ff (as below) in (3) one obtains:(as below) in (3) one obtains:

(1’(1’--2’) 2’)

(4) (4)

(5) (5)

••For For a givena given genusgenus of the surface, (of the surface, (55) gives the number of ) gives the number of ss--polygons. polygons. This condition is This condition is independent of the number of hexagonsindependent of the number of hexagons, which is , which is therefore therefore arbitraryarbitrary..••Special cases are the Special cases are the Platonic tilingsPlatonic tilings, with , with a single kind of polygonsa single kind of polygons, , and the and the Archimedean tilingsArchimedean tilings, with , with two different kinds of polygonstwo different kinds of polygons, one , one of which being here the hexagon.of which being here the hexagon.••In In Platonic fullerenesPlatonic fullerenes ((gg = 0): from (5), = 0): from (5), ff55 =12, =12, or or ff44 == 6 or 6 or ff33 == 4. 4. Archimedean fullerenes must always contain 12 Archimedean fullerenes must always contain 12 ff55; thus ; thus ff66 comes out comes out from (1from (1’’--22’’): ): 55ff55 + 6+ 6ff66 = 60+ 6= 60+ 6ff6 6 = 3= 3v v ; ; ff6 6 = (v/2)= (v/2)--1010..

∑= s sff

)1(12)6( gfs ss −=−∑

Fullerene countingFullerene counting

efsv s s 23 =⋅= ∑

2121

Spiral codeSpiral code•• Spiral conjecture:Spiral conjecture:11 the surface of every fullerene polyheron may the surface of every fullerene polyheron may

be unwound in at least one way as a continuous spiral strip obe unwound in at least one way as a continuous spiral strip of f edgeedge--sharing faces.sharing faces.

•• Spiral codeSpiral code implies the existence of aimplies the existence of a Hamiltonian pathHamiltonian path. . •• Spiral codeSpiral code is useful in:is useful in:

–– systematic nomenclaturesystematic nomenclature–– enumeration and construction of isomersenumeration and construction of isomersNonNon--spiralability,spiralability,22 in: in: -- fused triples or quadruples of pentagonsfused triples or quadruples of pentagons-- between 20<N<1000, 28 between 20<N<1000, 28 TT (with four IPT) 289 (with four IPT) 289 DD22, 61 , 61 DD33, and 58 C, and 58 C33or or

CC22, derived by truncation of fullerenes , derived by truncation of fullerenes TT..CC176176 and Cand C380380 ((TT) are unspiralable.) are unspiralable.

-- Le(Le(M))Le(Le(M)) andand Q(M)Q(M),, MM beingbeing unspiralableunspiralable

1.1. D. E. Manolopoulos, J. C. May and S. E. Down, Chem. Phys. Lett.D. E. Manolopoulos, J. C. May and S. E. Down, Chem. Phys. Lett., 1991, , 1991, 181, 105181, 105--111.111.

2. G. Brinkmann, P. W. Fowler and M. Yoshida, MATCH, 1998, 32. G. Brinkmann, P. W. Fowler and M. Yoshida, MATCH, 1998, 38, 78, 7--17.17.

2222

A capped nanotube we call here aA capped nanotube we call here a tubulenetubulene

NN Cap Cap SpiralSpiral sequence:sequence: ClassClass

66k k k k 66k k (56)(56)kk-- AA[2[2kk,,nn]] fafa --tubulenestubulenes

44k k k k 55k k 77k k (56)(56)kk-- AA[2[2kk,,nn]] ta ta --tubulenestubulenes

33k k k k 55kk-- ZZ[2[2kk,,nn]] tztz --tubulenestubulenes

1313k k /2/2 k k (56)(56)kk /2/2(665)(665)kk /2/2-- ZZ [3[3kk,,nn]] fzfz ––tubulenestubulenes

1111k k k k 66k k (56)(56)k k (65)(65)kk -- ZZ[2[2kk,,nn] ] kfkfzz ––tubulenestubulenes

99kk k k (56)(56)kk/2/2(665)(665)kk/2/2(656)(656)kk/2 /2 77kk-- ZZ [2[2kk,0] ,0] ((5,6,7)3) ((5,6,7)3) kfzkfz --tubulenestubulenes1212k k k k (56)(56)kk/2/2(665)(665)kk/2 /2 6633kk/2 /2 (656)(656)kk/2 /2 77kk-- ZZ [2[2kk,0] ,0] ((5,6,7)3) ((5,6,7)3) kfkfz z ––dvsdvs

1111k k k k 55k k 77k k 5522k k 77k k -- ZZ[2[2kk,,nn]] ((5,7)3) ((5,7)3) kfzkfz --tubulenestubulenes

Building ClassificationBuilding Classification