Fullerenes 1

Fullerenes 1
Mircea V. Diudea
Faculty of Chemistry and Chemical Engineering
Babes-Bolyai
Babes Bolyai University
400084 Cluj, Cluj,
ROMANIA
diudea@chem.ubbcluj.ro
1

Contents
• Fullerenes – Short History
• Basic Relations in Polyhedra
2

Fullerenes: Short History
(by Kroto 1 )
1. Kroto, Kroto,
H. The first predictions in the Buckminsterfullerene crystal ball. ball.
Fuller. Fuller. Sci. Sci.
Technol. Technol.
1994, 22, , 333-342. 333 342.
3

Before History
• The classic text of D’Arcy Thompson “ “On On
Growth Growth and and Form”, Form”,
Cambridge Univ. Press
(1942), speaks about an
Aulonia Aulonia hexagona,
hexagona
a sea creature with a silicious skeleton.
4

Aulonia Aulonia hexagona
hexagona (by Haeckel)
Haeckel
5

Before History
• Jones, D. E. H. New New Scientist, Scientist,
1966, 32, p. 245.;
• Jones, Jones,
D. E. H., The The Inventions Inventions of of Daedalus, Daedalus
Freeman: Oxford, 1982, pp. 118-119. 118 119.
“the the high temperature graphite production might
be modified to generate graphite balloons”.
balloons”.
6

First Considerations
• Eiji Osawa, Osawa Kagaku Kagaku (Kyoto) 1970, 25, 25,
854-863 854 863
(in in Japanese);
Japanese);
Chem. Chem.
Abstr. Abstr.
1971, 74, 74,
75698v.
(The original conjecture of a stable C 60 molecule).
Yoshida, Z.; Osawa, Osawa,
E. Aromaticity.
Aromaticity.
Kagakudojin: Kagakudojin:
Kyoto,
1971 (in Japanese)
Bochvar, Bochvar,
D. A.; Gal’pern, Gal’pern,
E. G. Dokl. Dokl.
Akad. Akad.
Nauk Nauk SSSR, SSSR, 1973,
209, 209,
610-612. 610 612. (English translation, Proc. Acad. Sci. Sci.
USSR,
1973, 209, 209,
239-241).
239 241).
(Huckel calculations)
7

Eiji Osawa and Mircea Diudea
Okazaki, Japan, Jan. 8, 2004
8

Calculations and Nomenclature
• Davidson, R. A. Theor
Theor. . Chim. Chim.
Acta, Acta,
1981, 58, 58,
193-195. 193 195.
(Huckel Huckel calculations)
• Castells, Castells,
J.; Serratosa, Serratosa,
F. J.,
941. (ibid. 1986, 63, 630)
(C 60 and C 60
• Haymet, Haymet,
A. D. J.,
60H 60
J., Chem. Ed., 1983, 60,
60 (IUPAC Nomenclature)
J., Chem. Chem. Phys Phys Lett., Lett.,
1985, 122, 122,
421- 421
424. (Huckel Huckel calculations)
9

C60 60 – First Syntheses
• Kroto, Kroto,
H.; Heath, J. R.; O’Brian, S. C.; Curl, R. F.;
Smalley, R. E. (Nobel Prize-1995)
Prize 1995)
Sussex University (UK) & Rice University (USA), ( USA),
Buckminsterfullerene C60 60 isolated from self-assembly
self assembly
products of graphite heated by plasma.
Nature Nature (London) , 1985, 318, 318,
162-163. 162 163.
• Kraetschmer, Kraetschmer,
W.; Lamb, L. D.; Fostiropoulos, Fostiropoulos,
K.;
Huffman, D. R., Solid C
Solid C60: 60:
a new form of carbon. C 60
isolated in macroscopic amount by arc vaporization of
graphite.
Nature Nature (London) , 1990, 347, 347,
354-358. 354 358.
10

Nanotubes and Tori
• S. Iijima, Iijima Helical microtubules of graphitic
carbon. Nature Nature (London), 1991, 354, 354,
56-58. 56 58.
• Liu, J.; Dai, H.; Hafner, Hafner,
J. H.; Colbert, D. T.;
Smalley, R. E.; Tans, S. J.; Dekker, Dekker,
C., Fullerene
"crop circles". Nature, Nature 1997, 385, 385,
780-781 780 781
• R. Martel, H. R. Shea, Shea,
and Ph. Avouris, Avouris Ring
formation in single-wall single wall carbon nanotubes.
nanotubes
J. J. Phys. Phys. Chem. Chem.
B, B, 1999 1999, , 103 103,, 7551-7556.
7551 7556.
11

Mircea Diudea and Sumio Iijima,
Okazaki, Japan, Jan. 8, 2004
12

Isolated Fullerenes
• N N = 20, 36, 60, 70, 76, 78, 82, 84, 96
• H. Prinzbach et al., Nature, 2000, 407, 60-63 60 63 / M. Saito and
Miyamoto,
Miyamoto,
Phys. Phys. Rev. Rev. Lett., Lett.,
2001, 87, 035503 / J. Lu et al., Phys. Phys.
Rev. Rev. B, B,
2003, 67, 125415.
• C. Piskoti, J. Yarger, and A. Zettl, Nature, Nature,
1998, 393, 771-773. 771 773.
• R. Ettl, I. Chao, F. Diederich, R. L. Whetten, Nature, Nature,
1991, 353, 353,
149.
• F. Diederich, R. L. Whetten, C. Thilgen, R. Ettl, I. Chao, and M.
M. Alvarez, Science, Science,
1991, 254, 254,
1768.
• K. Kikuchi, N. Nakahara, T. Wakabayashi, S. Suzuki, H.
Shiromaru, Y. Miyake, K. Saito, I. Ikemoto, M. Kainosho, and Y.
Achiba, Nature, Nature,
1992, 357, 357,
142.
13

Fullerene = Cage tiled with pentagons (12)
and hexagons (N/2-10) (N/2 10)
1812 topological isomers
C60 60 (II hh) (side)
C60 60 (N=12k; k = 5) (top)
14

C Nomenclature 60 Nomenclature
60
1,2
• Hentriacontacyclo[29.29.0.0
Hentriacontacyclo[29.29.0.0 2,14 .0 3,12 .0 4,59 .0 5,10 .0 6,5
8 .0 7,55 .0 8,53 .0 9,21 .0 11,20 .0 13,18 .0 15,30 .0 16,28 .0 17,25 .0 19,24 .
0 22,52 .0 23,50 .0 26,49 .0 27,47 .0 29,45 .0 32,44 .0 33,60 .
0 34,57 .0 35,43 .0 36,56 .0 37,41 .0 38,54 .0 39,51 .0 40,48 .
0 42,46
42,46 ] 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.
15

Basic Relations in Polyhedra
• First theorems on graph counting (Euler ( Euler) 1,2
∑d ( dv dvdd
) = 2 2ee (1)
∑s ( sf sfss
) = 2 2ee (2)
where vv dd and ff ss denote vertices of degree d d and ss--sized sized faces,
respectively.
1. Euler, L. Solutio Problematis ad Geometriam Situs Pertinentis.
Comment. Acad. Sci. I. Petropolitanae 1736, 8, , 128-140. 128 140.
2. King, King,
R. B., Applications of Graph Theory and Topology in
Inorganic Cluster and Coordination Chemistry, Chemistry,
CRC Press, 1993.
16

Planar Graph
A graph is planar planar if it can be drawn in the plane without crossings.
Its regions are called faces, faces,
ff. . The unbounded region is called
the exterior exterior face (Harary ( Harary). ). 1
A graph is planar if and only if it has no subgraphs homeomorphic
homeomorphic
to
(Kuratowski). 2
to KK 55 or KK 3,3 3,3 (Kuratowski).
f 1
f 2
f 3 f 4
1. Harary , F. Graph Theory, Theory,
Addison - Wesley, Reading, M.A., 1969.
2. Kuratowski, K. Sur la Problème des Courbes Gauches en Topologie,
Topologie,
Fund. Math. Math.
1930, 15, 15,
271-283. 271 283.
17

Euler Theorem on Polyhedra 1
v – e + f = χχ = 2(1 – g) (3)
χχ = Euler’s Euler s characteristic
v = number of vertices,
e = number of edges,
f = number of faces,
g = genus ; (g = 0 for a sphere; 1 for a torus). torus
A consequence:
consequence
A sphere can not be tessellated only by hexagons.
Fullerenes need 12 pentagons (for closing the cage) and (N/2-10) (N/2 10) hexagons. hexagons
In the opposite, a tube and a torus allow pure hexagonal nets.
1. L. Euler, Elementa doctrinae solidorum, Novi Comment. Acad. Sci. I. Petropolitanae
Comment. Acad. Sci. I. Petropolitanae
1758, 4, 109-140.
18

Fullerene counting
Rewrite relations (1), (2), and (3) as:
• 3vv = 2 2ee (1’)
•
•
5ff 5 + 6 6ff 6 = 2 2ee
v v + ff = 2 + ee
(2’)
(3’)
• Substituting v in (3’) by its value from (1’) one can write:
• (2/3) (2/3)ee + f f = 2 + ee
• 2ee + 3 3ff = 6 + 3 3ee
• ee = 3 3f f – 6 (4)
• Expressing ff by its composition: ( (ff = 5 ff + f 6), ), relation (4) becomes:
• ee = 3( 3(ff 5 + 6) ff ) – 6 (5)
• Substituting e from (5) in (2’) one obtains:
• 5ff 5 + 6 6ff 6 = 6( 6(ff 5 + 6) ff ) – 12
• 5 ff = 12 (6)
• From (1’), (2’), and (6) the expression for 6 ff is obtined:
• 5ff 5 + 6 6ff 6 = 3 3vv
• 6 ff = vv/2 /2 – 10 (7)
19

Fullerene counting
By substituting vv, , ee and ff (as below) in (3) one obtains:
3 v = ∑
s ⋅ f e
s s = 2
f
∑
=
s
∑
f
s s
s) f s
( 6 − = 12(
1−
g)
(1’-2’) (1’ 2’)
•For For a given genus of the surface, (5) ( ) gives the number of s-polygons. polygons.
This condition is independent of the number of hexagons, hexagons,
which is
therefore arbitrary. arbitrary
•Special Special cases are the Platonic Platonic tilings, tilings,
with a single kind of polygons, polygons,
and the Archimedean Archimedean tilings, tilings,
with two different kinds of polygons, polygons,
one
of which being here the hexagon.
•In In Platonic fullerenes (gg = 0): from (5), 5 ff =12, or
Archimedean fullerenes must always contain 12 5; ff thus
from (1’-2’): (1 ): 5ff 5 + 6 6ff 6 = 60+ 6 6 = 3 3v v ; 6 = (v/2)-10 (v/2) 10.
= 60+ 6ff 6
; ff 6
(4)
(5)
or 4 ff = 6 or 3 ff = 4.
; thus 6 ff comes out
20

Spiral code
• Spiral conjecture: 1 the surface of every fullerene polyheron may
be unwound in at least one way as a continuous spiral strip of of
edge-sharing edge sharing faces.
• Spiral code implies the existence of a Hamiltonian path. path.
• Spiral code is useful in:
– systematic nomenclature
– enumeration and construction of isomers
Non-spiralability,
Non spiralability, 2 in: - fused triples or quadruples of pentagons
- between 20

Building Classification
A capped nanotube we call here a tubulene
NN Cap Spiral sequence: Class
6k k k k 6k k (56) kk -- A[2 [2kk,nn] fa fa -tubulenes tubulenes
4k k k k 5k k 7k k (56) kk -- A[2 [2kk,nn] ta ta -tubulenes tubulenes
3k k k k 5kk -- Z[2 [2kk,nn] tz tz -tubulenes tubulenes
13 13k k /2 k k (56) kk /2 (665) kk /2 /2-- Z [3kk,nn] [3
fz fz –tubulenes tubulenes
11 11k k k k 6k k (56) k k (65) kk -- Z[2 [2kk,nn] ] kfzz kf –tubulenes tubulenes
9kk k k (56)
12 12k k k k (56)
(56) kk/2 /2 (665) kk/2 /2 (656) kk/2 /2 7kk -- Z [2kk,0] [2
(56) kk/2 /2 (665) kk/2 /2 63kk/2 /2 (656) kk/2 /2 7kk -- Z [2
,0] ((5,6,7)3) kfz kfz -tubulenes tubulenes
[2kk,0] ,0] ((5,6,7)3) kfz kfz
–dvs dvs
11 11k k k k 5k k 7k k 52k k 7k k -- Z[2 [2kk,nn] ((5,7)3) kfz kfz -tubulenes tubulenes
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