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From C60 to negatively curved graphite
Prog. Crystal Growth and Charact. Vol.34, pp. 25-36, 1997
Pergamon
© 1997 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 0960-8974/97 $32.00 PII: S 0 9 6 0 - 8 9 7 4 ( 9 7 | 0 0 0 0 3 - X
FROM C6o TO NEGATIVELY CURVED GRAPHITE Humberto Terrones* and Alan L. M a c k a y * ' Departmentof Chemistry, Universityof Cambridge, Lensfield Road, Cambridge CB2 1EW, U.K. * Department of Crystallography, BirkbeckCollege (Universityof London), Malet Street, London WC1E 7HX. U.K.
ABSTRACT With the discovery of C6o or Buckminsterfullerene a new kind of materials with important applications has emerged. Orginary graphite is composed of flat hexagonal layers. If pentagons are introduced, the graphite sheets start to curve in such a way that 12 pentagons are needed to close the structure and form fullerenes. Starting from the properties of 2D manifolds or surfaces, we have found that by introducing rings with more than six carbon atoms, periodic structures with the same topology as triply periodic minimal surfaces, we have found that by introducing rings with more than six carbon atoms, periodic structures with the same topology as triply periodic minimal surfacds can be constructed. The D, G, P, H and I-WP type surfaces have been obtained from graphite-like sheets. In terms of the Gaussian curvature K, ordinary and cylindrical graphite have K=O, Fullerenes have K>O and triply periodic surfaces decorated with graphite have K
KEYW'OBDS Flexicrystallography, periodic minimal surfaces, negative curvature INTRODUCTION In the past, the beauty of five fold and icosahedral symmetries was reserved just for mathematicians because of the small number of applications or examples known in Nature. Recently, important discoveries have shown us that icosahedral synunetry is present at different levels in the natural world; for example, in the structure of viruses, radiolaries, quasicrystals and C~o. This means that for some reasons Nature under certain circumstances prefers this symmetry. The revolution caused by the discovery of quasicrystals is a clear example that icosahedral symmetry plays an important role in metallic alloys (Shechtman et al., 1984). Unfortunately, until now there are not many applications for quasicrystals. However, their structure remains a mystery. Another remarkable discovery is Cs0 (Kroto et al., 1985), a truncated icosahedrom composed of 60 atoms of carbon. In this case, many applications have appeared and it seems that a new area in materials science and chemistry is being developed. Besides Cs0 and other fullerenes, other structures like cylinders and onion-like shapes have been found (lijima, 1991; Ugarte, 1992). In addition, spherical and cylindrical particles of tungsten disulphide have been reported by Tenne et al. (1992). In these structures the sheets 25
26
H. Terrones and A. L. Mackay
are curved by the introduction of "defects" (pentagonal rings). In this paper we show that curving structures in different ways lead to interesting possibilities which can have important applications. In the case of graphite, we have introduced rings with more than six atoms of carbon producing periodic saddle surfaces which have the same shapes of triply periodic minimal surfaces. We have found that there is agreat variety of these hypothetical structures and that these are more stable than Coo. The geometry and other properties wiU be given.
GEOMETRY OF C U R V E D G R A P H I T E 2-d M a n i f o l d s a n d t h e S t r u c t u r e of M a t e r i a l s : Flexi- c r y s t a l l o g r a p h y The mathematics of2-D manifolds is useful for proposing and characterising structures in which the atoms or units rest on surfaces with different curvatures. These curved arrangements of atoms have been called flexicrystals by Alan Mackay. Therefore, flexicrystallography gives an alternative way to understand complex structures in which the atomic planes do not meet at straight lines. Ordinary graphite is one example of a flat layered structure composed of hexagonal rings of carbon. Nature has shown us that not all structures are flat, but present curvature. The concept of curvature plays an important role in geometry allowing us to escape from traditional Euclidean to ellipsoidal and hyperbolic geometries. In simple terms, the curvature at a point can be defined as the inverse of the radius of the best fitting circle to a given curve at this point. If a surface (2-D manifold) is considered then two circles are needed, so two curvatures are associated with the point. When both curvatures are either maxima or minima they are called principal curvatures kl and k2. The Gaussian curvature K is the product of the two principal curvatures K = kl k2 and the mean curvature is H=kl+k2/2. It turns out that ordinary graphite can be bent by introducing or taking out atoms of the hexagonal rings. When a ring with five atoms is generated, a structure with positive Gaussian curvature is obtained. If the number of atoms is greater than six then the Gaussian curvature becomes negative (saddle shape). In Ceo and in other fullerenes 12 pentagons are needed to close the structure. With the introduction of rings with more than six atoms, periodic structures with topologies similar to triply periodic minimal surfaces can be obtained (TPMS). These ordered graphite foams present channels which resemble those seen in zeolites, and in some cases divide space into two congruent regions. Also cylindrical graphite can be produced by rolling up a flat layer of graphite, here the Gaussiazl curvature is zero, but one of the principal curvatures is the inverse of the radius of the cylinder. These cylinders have been observed experimentally (lijima, 1991). Graphite is just one example of an arrangement of atoms that call be bent. Recently Tenne et al. (1992) have reported spherical and cylindrical structures of tungsten disulphide, another layered compound. T r i p l y P e r i o d i c M i n i m a l Surfaces f r o m G r a p h i t e S h e e t s It has been mentioned in the previous section that graphite can take shapes which correspond to triply periodic minimal surfaces, but what is a minimal surface? A minimal surface is a surface which presents zero mean curvature at every point, so the Gaussian curvature is non-positive (saddle shape). A simple example of minimal surfaces are the soap films obtained when introducting a closed wire into soapy water. Since the mathematical point of fiew there are many open questions regarding the surfaces which minimize area and other energy functionals. Last century, Herman Amandus Schwarz found that patches of minimal surfaces could be accommodated smoothly to produce periodic minimal surfaces (Schwarz, 1890). To calculate the coordinates of minimal surfaces, a representation given by Karl Weierstrass (Professor of Schwartz) can be used. This representation involves the computation of contour integrals in the complex plane to generate the coordinates in real space (Weierstrass, 1866). Starting from the Weierstrass equations we have been able to decorate the D (or F), P, G and H TP MS with graphite sheets [7,13,14]. The negative curvature is due to octagonal rings of carbon (see figures 1-5). If heptagons are used then surfaces parallel to TPMS can be generated. In this case the structures do not divide space into two congruent regions (Lenosky et al., 1992; Vanderbilt
From Cso to Negatively Curved Graphite
27
and Tersoff, 1992).
Topology of Periodic 2-D Manifolds Decorated with A t o m s Assuming that the atoms rest on a periodic surface, the genus of the structure can be calculated using the Gauss-Bonnet theorem which relates the curvature to the topology. For a compact surface without boundaries the Gauss-Bonnet theorem can be written as
f Kds = 2r(2 - g) orientable surface f Kds = 2r(1 - g) non - orientable surface
(1)
where K is the Gaussian curvature and g the genus of the surface (number of handles). A surface is orientable when it has two sides and non-orientable when it is one sided like the Mobius strip. Since the structures considered are periodic, it is possible to compactified the primitive cell by identifying the opposite sides and glueing them with handles, so the structure is properly embedded in the three-torus (Hyde, 1989). Knowing the number of different carbon rings, the genus can be obtained from
N5 - N7 - 2N8 = 6X = 12(1 - g )
orientable surface
N5 - N7 - 2N8 = 6X = 6(1 - g) non - orientable surface
(2) (3)
Where N5, N7 and N8 are the number of pentagonal, heptagonal and octagonal rings, X is the Euler characteristic, and g the genus. Since the hexagonal rings do not contribute to the curvature of the structure, do not appear in equations 2 and 3. In the ease of Cs0 and closed fullerenes there are 12 pentagons, so the genus is zero as in a sphere. In the orientable primitive cells of the D, G and P TPMS there are 12 octagonal rings, therefore, g = 3. When these surfaces are constructed as non-orientable, the space groups are different and the number of octagonal rings per primitive cell is 6; in this case equation 3 must be used, giving again g = 3.
Mathematical Transformations and Kinematic Processes There are several mathematical transformations in surfaces which can help to understand physical changes. Among these transformations are the Bonnet (1853), Goursat, a mixture of these and transformations which invert the sides of the surface, by the Bonnet transformation it is possible to change the P into G and D TPMS preserving the metric, the mean and Gaussian curvatures. Therefore, bending without stretching takes place. The Goursat transformation changes the metric and curvatures, but preserves the angles, so it is conformal. We have been able to combine these two transformations showing that an interesting and realistic possibility involves stretching and bending (Terrones, 1992). Hyde and Andesson (1986) have proposed a mechanism based on the Bonnet transformation for the martensitie transition. On the other hand, with transformations operating on surfaces with constant mean curvature, it is possible to invert cylinders and spheres. This might be important in fullerenes and cylindrical graphite. Corannulene C20H10 is one example of a molecule which inverts itself 200,000 times per second (Bonnet, 1853). STABILITY OF CURVED GRAPHITE The stability of 11 negatively curved graphite structures has been measured by using a three-body potential introduced by Tersof [15.16]. The results indicate that these structures are more stable than C6o (see table I and figure 6). The main reason for high stability is that the 120 bond angles and lengths of ordinary graphite are ahnost preserved in the heptagonal and octagonal carbon rings
28
H. Terrones and A. L. Mackay
Figure 1: Triply Periodic Minimal Surfaces (TPMS) Decorated with Graphite: Four unit cells of the P surface with octagonal rings.
From Cso to Negatively Curved Graphite
Figure 2: Triply Periodic Minimal Surfaces (TPMS) Decorated with Graphite: unit cell of the G surface with octagonal rings.
29
30
H. Terrones and A L. Mackay
s
~ p ~q ~
J
¢,
s
"1), "
~
~
t~
Figure 3: Triply Periodic .Minimal Surfaces ,"]'I)_NI%IDecorated with Graphite: unit cell of the D surface with octagonal rings.
m
From Cso to Negatively Curved Graphite
Figure 4: The I-WP graphite structure with pentagonal and octagonal rings
31
H. Ten'ones and A. L. Mackay
32
Table 1: Stability of curved graphite: Energy relative to graphite in eV.
Surface
Energy eV
Author
P8 G8 D8 P7 G7 D7 P688 G688 D688 D766 I- WP C60 C20
0.2228 0.2494 0.2830 0.4711 0.1760 0.2774 0.5297 0.2359 0.0449 0.1407 0.5701 0.6658 1.6166
Mackay & Terrones (1991,1992a,b) Mackay & Terrones (1991,1992a,b) Mackay,& Terrones (1991,1992a,b) Lenosky et al. (1992) Lenosky et al. (1992) Lenosky et al. (1992) O'Keeffe et al. (1992) Mackay & Terrones ( 1991,1992a,b) O'Keeffe et al. (1992) Vanderbit & Tersoff (1992) Mackay & Terrones (1991,1992a,b) Kroto et al. (1985)
needed to construct the periodic structures. In the pentagonal rings of Cs0 the bond angles are 108. Another reason for the stability is the infinitely periodic character of negatively curved graphite since dangling bonds can be eliminated by adding cells, so every atom has always three neighbours. In other words, having open ends costs energy and cause instability. The potential given by Tersoff can be written as
E = l ~ Vii = fc(r'ij)[aijfn(ij) + biJA(rq)] 2 i#j
(4)
Where E is the total energy, Vij is the bond energy, rij is the distance between atom i and atom j, and
fn(r) = A e -A'r fA(r) = B e - ~ 1, 1/2 - 1/2sin(rr(r - R ) / 2 D ) 0,
fa(r) =
if r i R-D } R-D i r i R + D r/, R+D
~j = Ek#i,, f,-(ri,k g(eljk) exp[A~ (rii - rik) 3 ] g(0) = 1 + c2/d 2- c a /(d 2 + ( h - cos0) 2 Where 0ijk is the bond angle between bonds ij and ik. a,~ = (1 +
r]ij = E k # i j fc(rl,k exp[A~ (r O
- rlk) 3
c~",f)) -1/(2"o
]
The parameters for carbon are : A = 1393.6 eV, B = 346.74 eV, 3,1 = 3.4879 A, A2 --- 2.2119 A, = 1.5724 x 10-7, n = 0.72751, c = 38049, d = 4.3484, - 0.570.58, R = 1.95A, D = 0.15A, a = 0 and A3 = 0.
From C60 to Negatively Curved Graphite
33
Figure 5: The G688 graphite structure with octagonal rings
Table 2: Properties of D, G, P, I-WP and G688 surfaces made with graphite. a is the lattice parameter in A, Np is the number of atoms per primitive rombohedral cell. Nc is the number of atoms per cubic unit cell. 9p is the genus per primitive cell and p is the density in g r / m 3 Surface a Np Arc gp p Symmetry
D surface 24.09 192 768 3 1.096 Pn3m - Fd3m
G surface 18.98 192 384 3 1.129 l a 3 d - I4132
P surface 15.41 192 192 3 1.049
I-WP 24.09 372 744 4 1.061
G688 9.5 48 96 3 2.21
Irn3m - Pm3m
ITt3m
la3d
34
H. Terrones and A. L. Mackay
Table 3: Properties of P688, D688, P7 and D766 surfaces made with graphite. a is the lattice parameter in A, N~ is the number of atoms per primitive rombohedral cell. N~ is the number of atoms per cubic unit cell. gp is the genus per primitive cell and p is the density in g r / m 3 Surface a
Np Nc g~ p Symmetry
E
P688 sudace 7.770 24 48 3 2.04
D688 surface 6.033 24 24 3 2.19
P7 surface 16.2 216 216 3 1.013
Im3m
Pn3m
Prelim
D766 21.8 168 672 3 1.28 Fd3
eV)
1.5
1.25
0.75
I°
ly6o P6 8 8
R
0.5
I-WP
0.25
2
4
6
8
i0
12
Figure 6: Potential energy per atom E in eV (relative to graphite) for different forms of curved graphite. The solid line indicates the energy of Coo. For ordinary graphite E=0
From Cso to Negatively Curved Graphite
35
CONCLUSION
It has been shown that interesting structures with different symmetries can be obtained by curving fiat sheets. Positive Gaussian curvature is due to the introduction of pentagonal rings and negative Gaussian curvature by using rings with more than six atoms. With the negative curvature, periodic structures with the same topology as triply periodic minimal surfaces have been constructed. With positive curvature closed shells like fullerenes are generated. According to a three-body potential proposed by Tersoff, periodic graphite foams are more stable than Ce0. It looks that graphite and tungsten disulphide axe the first examples of fiat arrangements of atoms that can be curved and should be more waiting to be discovered. ACKNOWLEDGEMENTS
One of us (H.T.) is very grateful to Dr. Jaeek Klinowski of the University of Cambridge for his kind invitation to collaborate in his group, and Jose Luis Aragon of the Institute of Physics UNAM, Mexico for allowing the use of computing facilities to finish this manuscript. REFERENCES
Bonnet, O. (1853). Note sur la theorie generale des surfaces, C.R. Acad. Sci. Paris, 37, 529-532. Hyde, S.T. and Andersson, S. (1986). Kristallogr., 174, 225-236.
The martensite transition and differential geometry, Z.f.
Hyde, S.T. (1989). The topology and geometry of infinite periodic surfaces, Z.f. Kristallogr., 187, 165-185. Iijima, S. (1991). Helical Mieroturbules of Graphitic Carbon, Nature, 354, 56-58. Kroto, H.W., Heath, J.R., O'Brien, S.C., Curl, R.F. and Smalley, R.E. (1985) Cs0 : Buckminsterfullerene, Nature, 318,162- 163. Lenosky T., ~onze X., Teter M. and Elser V. (1992). Energetics of Negatively Curved Graphitic Carbon, Nature, 355, 333-335. Mackay, A.L., and Terrones H. (1991). Diamond from Graphite, Nature 352, 762. O'Keeffe, M., Adams, G.B. and Sankey, O.F. (1992). Predicted New Low Energy forms of Carbon, Phys. Rev. Let., 68, No. 15, 2325- 2328. Schwarz H.A. (1890). Gesammelte Mathematische Abhandlungen, vol.1, Springer, Berlin. Shechtman, D.S., Bleeh, I., Gratias, D. and Cahn, J.W. (1984). Metallic Phase with Long Range Orientational Order and No Traslational Symmetry, Phys. Rev. Lett., 53, 1951-1953. Scott, T.L., Hashemi, M.M. and Bratcher, M.S. (1992). Corannulene Bow-to-Bowl inversion is Rapid at Room Temperature, J.Am.Chem.Soc., 114, 1920-1921. Tenne, R., Margulis, M., Genut, M., and Hodes, G. (1992). Polyhedral and cylindrical structures of tungsten disulphide, Nature, 360, 444-446. Terrones, H. (1992). Mathematical Surface and Invariants in the Study of Atomic structures, Ph.D., Thesis, University of London. Terrones, H., and Mackay, A.L. (1992). The Geometry of Hypothetical Curved Graphite Structures, Carbon, 30 Nos. 8, 1251-1260. Tersoff, J. (1988). New Empirical Approach for the Structure and Energy of Covalent Systems, Phys. Rev. B, 37, 6991-7000. Tersoff, J. (1988). Empirical lnteratomie Potential for Carbon, with Applications to Amorphous Carbon. Phys. Rev. 61, No. 265, 2879-2882).
36
H. Ten~nes and A, L. Mackay
Ugarte, D. (1992). Curling and Closure of Graphitic Networks under Electron-beam irradiation, Nature, 359, 707-709. Vanderbilt D., Tersoff J. (1992). Negative-Curvature Fullerene Analog of Ceo, Phys. Rev. Lett., 68, 511-513. Weierstrass, K. (1866). Untersuchungen uber die Flachen, deren mittlere Krummung uberall gleich null ist, Monatsber, d. Berliner Akad, 612-625.
Pergamon
© 1997 Published by Elsevier Science Ltd Printed in Great Britain. All rights reserved 0960-8974/97 $32.00 PII: S 0 9 6 0 - 8 9 7 4 ( 9 7 | 0 0 0 0 3 - X
FROM C6o TO NEGATIVELY CURVED GRAPHITE Humberto Terrones* and Alan L. M a c k a y * ' Departmentof Chemistry, Universityof Cambridge, Lensfield Road, Cambridge CB2 1EW, U.K. * Department of Crystallography, BirkbeckCollege (Universityof London), Malet Street, London WC1E 7HX. U.K.
ABSTRACT With the discovery of C6o or Buckminsterfullerene a new kind of materials with important applications has emerged. Orginary graphite is composed of flat hexagonal layers. If pentagons are introduced, the graphite sheets start to curve in such a way that 12 pentagons are needed to close the structure and form fullerenes. Starting from the properties of 2D manifolds or surfaces, we have found that by introducing rings with more than six carbon atoms, periodic structures with the same topology as triply periodic minimal surfaces, we have found that by introducing rings with more than six carbon atoms, periodic structures with the same topology as triply periodic minimal surfacds can be constructed. The D, G, P, H and I-WP type surfaces have been obtained from graphite-like sheets. In terms of the Gaussian curvature K, ordinary and cylindrical graphite have K=O, Fullerenes have K>O and triply periodic surfaces decorated with graphite have K
KEYW'OBDS Flexicrystallography, periodic minimal surfaces, negative curvature INTRODUCTION In the past, the beauty of five fold and icosahedral symmetries was reserved just for mathematicians because of the small number of applications or examples known in Nature. Recently, important discoveries have shown us that icosahedral synunetry is present at different levels in the natural world; for example, in the structure of viruses, radiolaries, quasicrystals and C~o. This means that for some reasons Nature under certain circumstances prefers this symmetry. The revolution caused by the discovery of quasicrystals is a clear example that icosahedral symmetry plays an important role in metallic alloys (Shechtman et al., 1984). Unfortunately, until now there are not many applications for quasicrystals. However, their structure remains a mystery. Another remarkable discovery is Cs0 (Kroto et al., 1985), a truncated icosahedrom composed of 60 atoms of carbon. In this case, many applications have appeared and it seems that a new area in materials science and chemistry is being developed. Besides Cs0 and other fullerenes, other structures like cylinders and onion-like shapes have been found (lijima, 1991; Ugarte, 1992). In addition, spherical and cylindrical particles of tungsten disulphide have been reported by Tenne et al. (1992). In these structures the sheets 25
26
H. Terrones and A. L. Mackay
are curved by the introduction of "defects" (pentagonal rings). In this paper we show that curving structures in different ways lead to interesting possibilities which can have important applications. In the case of graphite, we have introduced rings with more than six atoms of carbon producing periodic saddle surfaces which have the same shapes of triply periodic minimal surfaces. We have found that there is agreat variety of these hypothetical structures and that these are more stable than Coo. The geometry and other properties wiU be given.
GEOMETRY OF C U R V E D G R A P H I T E 2-d M a n i f o l d s a n d t h e S t r u c t u r e of M a t e r i a l s : Flexi- c r y s t a l l o g r a p h y The mathematics of2-D manifolds is useful for proposing and characterising structures in which the atoms or units rest on surfaces with different curvatures. These curved arrangements of atoms have been called flexicrystals by Alan Mackay. Therefore, flexicrystallography gives an alternative way to understand complex structures in which the atomic planes do not meet at straight lines. Ordinary graphite is one example of a flat layered structure composed of hexagonal rings of carbon. Nature has shown us that not all structures are flat, but present curvature. The concept of curvature plays an important role in geometry allowing us to escape from traditional Euclidean to ellipsoidal and hyperbolic geometries. In simple terms, the curvature at a point can be defined as the inverse of the radius of the best fitting circle to a given curve at this point. If a surface (2-D manifold) is considered then two circles are needed, so two curvatures are associated with the point. When both curvatures are either maxima or minima they are called principal curvatures kl and k2. The Gaussian curvature K is the product of the two principal curvatures K = kl k2 and the mean curvature is H=kl+k2/2. It turns out that ordinary graphite can be bent by introducing or taking out atoms of the hexagonal rings. When a ring with five atoms is generated, a structure with positive Gaussian curvature is obtained. If the number of atoms is greater than six then the Gaussian curvature becomes negative (saddle shape). In Ceo and in other fullerenes 12 pentagons are needed to close the structure. With the introduction of rings with more than six atoms, periodic structures with topologies similar to triply periodic minimal surfaces can be obtained (TPMS). These ordered graphite foams present channels which resemble those seen in zeolites, and in some cases divide space into two congruent regions. Also cylindrical graphite can be produced by rolling up a flat layer of graphite, here the Gaussiazl curvature is zero, but one of the principal curvatures is the inverse of the radius of the cylinder. These cylinders have been observed experimentally (lijima, 1991). Graphite is just one example of an arrangement of atoms that call be bent. Recently Tenne et al. (1992) have reported spherical and cylindrical structures of tungsten disulphide, another layered compound. T r i p l y P e r i o d i c M i n i m a l Surfaces f r o m G r a p h i t e S h e e t s It has been mentioned in the previous section that graphite can take shapes which correspond to triply periodic minimal surfaces, but what is a minimal surface? A minimal surface is a surface which presents zero mean curvature at every point, so the Gaussian curvature is non-positive (saddle shape). A simple example of minimal surfaces are the soap films obtained when introducting a closed wire into soapy water. Since the mathematical point of fiew there are many open questions regarding the surfaces which minimize area and other energy functionals. Last century, Herman Amandus Schwarz found that patches of minimal surfaces could be accommodated smoothly to produce periodic minimal surfaces (Schwarz, 1890). To calculate the coordinates of minimal surfaces, a representation given by Karl Weierstrass (Professor of Schwartz) can be used. This representation involves the computation of contour integrals in the complex plane to generate the coordinates in real space (Weierstrass, 1866). Starting from the Weierstrass equations we have been able to decorate the D (or F), P, G and H TP MS with graphite sheets [7,13,14]. The negative curvature is due to octagonal rings of carbon (see figures 1-5). If heptagons are used then surfaces parallel to TPMS can be generated. In this case the structures do not divide space into two congruent regions (Lenosky et al., 1992; Vanderbilt
From Cso to Negatively Curved Graphite
27
and Tersoff, 1992).
Topology of Periodic 2-D Manifolds Decorated with A t o m s Assuming that the atoms rest on a periodic surface, the genus of the structure can be calculated using the Gauss-Bonnet theorem which relates the curvature to the topology. For a compact surface without boundaries the Gauss-Bonnet theorem can be written as
f Kds = 2r(2 - g) orientable surface f Kds = 2r(1 - g) non - orientable surface
(1)
where K is the Gaussian curvature and g the genus of the surface (number of handles). A surface is orientable when it has two sides and non-orientable when it is one sided like the Mobius strip. Since the structures considered are periodic, it is possible to compactified the primitive cell by identifying the opposite sides and glueing them with handles, so the structure is properly embedded in the three-torus (Hyde, 1989). Knowing the number of different carbon rings, the genus can be obtained from
N5 - N7 - 2N8 = 6X = 12(1 - g )
orientable surface
N5 - N7 - 2N8 = 6X = 6(1 - g) non - orientable surface
(2) (3)
Where N5, N7 and N8 are the number of pentagonal, heptagonal and octagonal rings, X is the Euler characteristic, and g the genus. Since the hexagonal rings do not contribute to the curvature of the structure, do not appear in equations 2 and 3. In the ease of Cs0 and closed fullerenes there are 12 pentagons, so the genus is zero as in a sphere. In the orientable primitive cells of the D, G and P TPMS there are 12 octagonal rings, therefore, g = 3. When these surfaces are constructed as non-orientable, the space groups are different and the number of octagonal rings per primitive cell is 6; in this case equation 3 must be used, giving again g = 3.
Mathematical Transformations and Kinematic Processes There are several mathematical transformations in surfaces which can help to understand physical changes. Among these transformations are the Bonnet (1853), Goursat, a mixture of these and transformations which invert the sides of the surface, by the Bonnet transformation it is possible to change the P into G and D TPMS preserving the metric, the mean and Gaussian curvatures. Therefore, bending without stretching takes place. The Goursat transformation changes the metric and curvatures, but preserves the angles, so it is conformal. We have been able to combine these two transformations showing that an interesting and realistic possibility involves stretching and bending (Terrones, 1992). Hyde and Andesson (1986) have proposed a mechanism based on the Bonnet transformation for the martensitie transition. On the other hand, with transformations operating on surfaces with constant mean curvature, it is possible to invert cylinders and spheres. This might be important in fullerenes and cylindrical graphite. Corannulene C20H10 is one example of a molecule which inverts itself 200,000 times per second (Bonnet, 1853). STABILITY OF CURVED GRAPHITE The stability of 11 negatively curved graphite structures has been measured by using a three-body potential introduced by Tersof [15.16]. The results indicate that these structures are more stable than C6o (see table I and figure 6). The main reason for high stability is that the 120 bond angles and lengths of ordinary graphite are ahnost preserved in the heptagonal and octagonal carbon rings
28
H. Terrones and A. L. Mackay
Figure 1: Triply Periodic Minimal Surfaces (TPMS) Decorated with Graphite: Four unit cells of the P surface with octagonal rings.
From Cso to Negatively Curved Graphite
Figure 2: Triply Periodic Minimal Surfaces (TPMS) Decorated with Graphite: unit cell of the G surface with octagonal rings.
29
30
H. Terrones and A L. Mackay
s
~ p ~q ~
J
¢,
s
"1), "
~
~
t~
Figure 3: Triply Periodic .Minimal Surfaces ,"]'I)_NI%IDecorated with Graphite: unit cell of the D surface with octagonal rings.
m
From Cso to Negatively Curved Graphite
Figure 4: The I-WP graphite structure with pentagonal and octagonal rings
31
H. Ten'ones and A. L. Mackay
32
Table 1: Stability of curved graphite: Energy relative to graphite in eV.
Surface
Energy eV
Author
P8 G8 D8 P7 G7 D7 P688 G688 D688 D766 I- WP C60 C20
0.2228 0.2494 0.2830 0.4711 0.1760 0.2774 0.5297 0.2359 0.0449 0.1407 0.5701 0.6658 1.6166
Mackay & Terrones (1991,1992a,b) Mackay & Terrones (1991,1992a,b) Mackay,& Terrones (1991,1992a,b) Lenosky et al. (1992) Lenosky et al. (1992) Lenosky et al. (1992) O'Keeffe et al. (1992) Mackay & Terrones ( 1991,1992a,b) O'Keeffe et al. (1992) Vanderbit & Tersoff (1992) Mackay & Terrones (1991,1992a,b) Kroto et al. (1985)
needed to construct the periodic structures. In the pentagonal rings of Cs0 the bond angles are 108. Another reason for the stability is the infinitely periodic character of negatively curved graphite since dangling bonds can be eliminated by adding cells, so every atom has always three neighbours. In other words, having open ends costs energy and cause instability. The potential given by Tersoff can be written as
E = l ~ Vii = fc(r'ij)[aijfn(ij) + biJA(rq)] 2 i#j
(4)
Where E is the total energy, Vij is the bond energy, rij is the distance between atom i and atom j, and
fn(r) = A e -A'r fA(r) = B e - ~ 1, 1/2 - 1/2sin(rr(r - R ) / 2 D ) 0,
fa(r) =
if r i R-D } R-D i r i R + D r/, R+D
~j = Ek#i,, f,-(ri,k g(eljk) exp[A~ (rii - rik) 3 ] g(0) = 1 + c2/d 2- c a /(d 2 + ( h - cos0) 2 Where 0ijk is the bond angle between bonds ij and ik. a,~ = (1 +
r]ij = E k # i j fc(rl,k exp[A~ (r O
- rlk) 3
c~",f)) -1/(2"o
]
The parameters for carbon are : A = 1393.6 eV, B = 346.74 eV, 3,1 = 3.4879 A, A2 --- 2.2119 A, = 1.5724 x 10-7, n = 0.72751, c = 38049, d = 4.3484, - 0.570.58, R = 1.95A, D = 0.15A, a = 0 and A3 = 0.
From C60 to Negatively Curved Graphite
33
Figure 5: The G688 graphite structure with octagonal rings
Table 2: Properties of D, G, P, I-WP and G688 surfaces made with graphite. a is the lattice parameter in A, Np is the number of atoms per primitive rombohedral cell. Nc is the number of atoms per cubic unit cell. 9p is the genus per primitive cell and p is the density in g r / m 3 Surface a Np Arc gp p Symmetry
D surface 24.09 192 768 3 1.096 Pn3m - Fd3m
G surface 18.98 192 384 3 1.129 l a 3 d - I4132
P surface 15.41 192 192 3 1.049
I-WP 24.09 372 744 4 1.061
G688 9.5 48 96 3 2.21
Irn3m - Pm3m
ITt3m
la3d
34
H. Terrones and A. L. Mackay
Table 3: Properties of P688, D688, P7 and D766 surfaces made with graphite. a is the lattice parameter in A, N~ is the number of atoms per primitive rombohedral cell. N~ is the number of atoms per cubic unit cell. gp is the genus per primitive cell and p is the density in g r / m 3 Surface a
Np Nc g~ p Symmetry
E
P688 sudace 7.770 24 48 3 2.04
D688 surface 6.033 24 24 3 2.19
P7 surface 16.2 216 216 3 1.013
Im3m
Pn3m
Prelim
D766 21.8 168 672 3 1.28 Fd3
eV)
1.5
1.25
0.75
I°
ly6o P6 8 8
R
0.5
I-WP
0.25
2
4
6
8
i0
12
Figure 6: Potential energy per atom E in eV (relative to graphite) for different forms of curved graphite. The solid line indicates the energy of Coo. For ordinary graphite E=0
From Cso to Negatively Curved Graphite
35
CONCLUSION
It has been shown that interesting structures with different symmetries can be obtained by curving fiat sheets. Positive Gaussian curvature is due to the introduction of pentagonal rings and negative Gaussian curvature by using rings with more than six atoms. With the negative curvature, periodic structures with the same topology as triply periodic minimal surfaces have been constructed. With positive curvature closed shells like fullerenes are generated. According to a three-body potential proposed by Tersoff, periodic graphite foams are more stable than Ce0. It looks that graphite and tungsten disulphide axe the first examples of fiat arrangements of atoms that can be curved and should be more waiting to be discovered. ACKNOWLEDGEMENTS
One of us (H.T.) is very grateful to Dr. Jaeek Klinowski of the University of Cambridge for his kind invitation to collaborate in his group, and Jose Luis Aragon of the Institute of Physics UNAM, Mexico for allowing the use of computing facilities to finish this manuscript. REFERENCES
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36
H. Ten~nes and A, L. Mackay
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