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Journal of MOLECULAR STRUCTURE

Journal of Molecular Structure 436-437 (1997) 1 I - 18

Investigations of diamond-graphite hybrids and fullerenes with seven-membered rings’

Rahul Sen, R. Sumathy, B.C. Satishkumar, C.N.R. Rae*

Solid Stute and Structural Chemistry Unit and Materials Research Centre, Indian Institute of Science, Bangalore 560012, Indiu. and Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore 560064, India

Received 21 April 1997; accepted 12 May 1997

Abstract

The nature of diamond-graphite hybrids has been studied by molecular mechanics, by examining the structures of species such as Cs4Hr wherein the sp3 to sp’ carbon ratio is varied progressively. The dependence of the average coordination number on the atom fraction of hydrogen has been examined in the light of the random covalent network model. The HOMO-LUMO gap has been estimated in a graphite-like Cl 10H 1 and in a diamond-like C lzoH r as a function of the sp”/sp* carbon atom ratio. The gap increases exponentially with the fraction of sp3 carbon. Shapes of fullerene-like structures with 7-membered rings, in addition to 6- and Smembered rings, have been investigated along with structures of bent nanotubes having similar ring systems. 0 1997 Elsevier Science B.V.

Keywords: Diamond-graphite hybrids; Fullerenes; Molecular mechanics

1. Introduction

There has been increasing interest in carbon structures in recent years, particularly because of the discovery of fullerenes and related forms of carbon containing five- and six-membered rings, with the hybridization of carbon between sp2 and sp” and with carbon-carbon bond distances between those of single and double bonds [l]. Another important aspect of carbon relates to hybrid structures between graphite and diamond, involving mixtures of sp2 and sp’ hybridized carbon atoms [2-41. In addition, several forms of carbon structures have been proposed

* Corresponding author. ’ Dedicated to Professor Henryk Ratajczak, a dear friend and an

outstanding structural chemist.

recently on the basis of theoretical considerations. Thus, on the basis of topological considerations, schwarzites involving 5, 6-, 7-, 8- and 9-membered rings have been suggested by Townsend et al. [5]. Hoffmann et al. [6] have proposed a possible new form of conducting carbon. We have studied the stability of structures of diamond-graphite hybrids with different sp2/sp’ atom ratios by employing the force field method. In addition to visualizing the hybrid structures, we have tried to understand their structures in terms of the dependence of the average coordination number as well as the sp’/sp’ atom ratio on the hydrogen atom fraction, xH, the hydrogen atoms being present on the surface and the edges of diamond-type structures. Some of the amorphous carbons are known to contain 20-60% hydrogen, the percentage varying with the sp”/sp’ ratio [7,8].

0022.2860/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved PII SOO22-2860(97)00224-X

12 R. Sen et d/Journal of Molecular Structure 436-437 (1997) 1 l-18

We have made use of the random covalent network model [3] to examine the structural changes with xn. For this purpose, we have started with graphitic-type sheets containing only sp2 carbons and examined the effect of progressively introducing sp” carbons. We have also studied the effect of introducing sp2 carbons in diamond-like sp” networks.

It is of importance to know whether diamond- graphite hybrids exhibit a range of band gaps between those of graphite and diamond. Balaban et al. [9] have theoretically examined diamond layers connected by graphite strips and have predicted that the band gap should decrease with increase in the strip width. Band gap estimates for diamond-graphite hybrid structures with different fractions of sp3 carbons which take into account several possible alternatives are, however, not available. We have carried out a systematic study of the HOMO-LIMO gaps in a variety of diamond- graphite hybrids, with varying sp”/sp’ ratios, starting with a pure sp’ layer at the graphite end and a three dimensional sp3 network at the diamond end.

Besides diamond-graphite hybrid structures, we have investigated fullerenes and nanotubes containing seven-membered rings by employing force-field calculations. Fullerenes containing five-, six- and seven-membered rings provide synthetic challenges and we describe some of the simplest fullerenes of this category. The presence of seven-membered rings in nanotubes, giving rise to bent nanotubes, has indeed been observed [lo]. We have compared force-field minimized structures of such namotubes with observed transmission electron microscopic (TEM) and atomic force microscopic (AFM) images.

2. Method of study

In order to visualize the alternative structures of different hydrocarbons of the general formula Cs4H,, C , ,oH, and C tZOHx, we have carried out energy minimization using the molecular simulation program DISCOVER (Biosym Technologies, San Diego) and obtained the actual structures by using Insight II. This molecular mechanics program solves for the electronic wavefunction by using the force field approach and focuses on obtaining the geometries and their static energies. The quasi- Newton-Raphson (VA09A) algorithm coupled with

the Broyden-Fletcher-Goldfarb and Shanno (BFGS) method for Hessian updating is used for the minimization. The force field calculations are much faster than ab initio or semiempirical quantum chemical methods and describe the structures and energies of non-polar systems fairly satisfactorily [ 11,121. We have not given much credence to the actual energy values of the different hydrocarbon structures, but have used the relative energies of alternative structures of the same system to identify the likely stable structure.

We have calculated the average coordination number, m, in structures such as those in Fig. 1, from Eq. (1) by employing the relation

m = (3 - 2~) + (4 - 3xn)(NsP? lNs,z )/ ( I+ U&J /Nsp~ ,)

(1) where (N,,z/N,,z) is the ratio of sp3 to sp2 hybridized carbon atoms. We compare the m values so obtained with those predicted by Eq. (2).

m = (20 - 8xn)/7 (2)

While Eq. (1) gives the observed m value by simple counting, Eq. (2) of Angus and Jansen [3] gives the m value obtained by applying constraint theory to the random covalent network. A random covalent net- work is completely constraint when the number of constraints per atom is equal to the number of mechanical degrees of freedom per atom [ 13,141, a completely constraint network being one where the stabilization due to the bond energy is just balanced

(4

Of)

(e)

Fig. 1. Some possible structures of the general composition Cs4H,: (a) Cg4H2s (1.2% sp3 carbons); (b) CgbH70 (7.1% sp’ carbons); (c) and (c’) Cs4H3, (15.5% sp’ carbons); (d) and (d’) Cs4Hd6 (26.2% sp’ carbons).

R. Sm et al./Jmunal qf‘h4olrculur Structure 436-437 (1997) II-18

by the destabilization due to the strain energy. The number of constraints of an atom is the number of directed covalent bonds with its nearest neighbours, in other words its coordination number. Increasing the coordination number (covalent bonds) of an atom contributes to stabilization due to bond energy, but also causes destabilization due to strain energy. The optimal coordination number is one that just uses up the allowed degrees of freedom to balance these effects. The number of degrees of freedom is equal to the dimensionality (three). Thus. the coordination number of an atom or the average coordination number of the system can be derived using the constraint theory.

In order to study the effect of incorporating sp’ (or sp’) carbons in an sp’ (or sp”) network, on the band (HOMO-LUMO) gap, we have started with large hydrocarbons containing either only sp’ or sp” carbons and estimated the gaps by using extended Huckel theory (EHT) [ 151. It may be noted that, for a series of related systems, the EHT method provides estimates of HOMO-LUMO gaps which follow the trend in experimental values [15,16]. The atomic coordinates for the structures were obtained from force field minimization by using the molecular simulation program DISCOVER described earlier. Extended Huckel calculations were performed by using the ICON version 8 (by R. Hoffmann) and were run in a VAX-88 machine. The program uses parametrized basis sets consisting of H 1s and C 2s and 2p orbitals for hydrocarbon molecules.

Fullerene-type structures having pentagons, hexagons and heptagons were built following Euler’s theorem:

L’ - t’ +,f = k (3)

where 11, e and f are the number of vertices, edges and faces of a polyhedral object, and k is the Euler characteristic and is related to the number of holes, g, according to the relation k = 2( 1 - g). g equals zero for closed-cage structures like fullerenes having no holes. In fullerenes three edges extend from each vertex. hence

3r=2e (4)

If there aref,, faces of n-membered rings, then

f-CA, (3

and

2e= 2 (6)

Eqs. (3)-(6) give rise to

for fullerenes containing pentagons, hexagons and heptagons,

J%--f7=12

We can therefore build closed-cage fullerene structures when the number of pentagons minus the number of heptagons equals twelve. All nanotube and fullerene-like structures were built and energy minimized using the force-field approach in the molecular simulation program DISCOVER.

13

3. Results and discussion

3.1. Structural studies of diamond-graphite hybrids

We have first studied the effect of increasing the number of sp3 carbons in a graphite-like system with the general formula C*“H,. In Fig. 1 we show typical structures of hydrogenated carbons with the com- position varying anywhere between Cx.,HZs and CgdHhh wherein, with increasing proportion of sp3 carbons, the fraction of hydrogen atoms, xn, also increases. Thus, CgjHIS has one sp” carbon, while CgaHhh has 42 sp’ carbons. In some of the com- positions, especially those with small xn, we have considered alternative structures. For example, in C84Hj” with six spi carbons we have considered three alternative structures, the structure shown in Fig. lb being the most stable. The structures (c) and (c’) as well as (d) and (d’) in Fig. 1 (containing 13 and 22 sp3 carbons respectively) are alternative structures. With the increase in xn or the number of sp’ carbons, the number of alternative structures with random arrangements decreases significantly.

Diamond-like carbons (DLCs) generally contain 10-25s sp’ carbons [2]. Although there has been some speculation on the structures of DLCs, very few studies have been carried out on them. We have

14 R. Sen et al./Journal of Molecular Structure 436-437 (1997) 11-18

(d)

(W

(d”)

Fig. 2. Some possible structures of diamond-like carbons of composition CUHx: (a) Cs4HY6 (2.4% sp* carbons); (b), (b’) and (b”) Cs4Hs0 (7.1% sp* carbons); (c) and (c’) Cs4Hs4 (11.9% sp’ carbons); (d) and (d’) Cs4Hsb (14.3% sp’ carbons).

considered a large number of structures with the general formula CsdH,, with the percentage of sp* carbons varying in the range of 2-30%. In Fig. 2, we show a few typical structures with varying num- bers of C=C bonds. Among the structures containing three double bonds (Fig. 2(b), (b’) and (b”)), 2(b”) with a benzene ring is considerably more stable (by 440-560 kJ) than the structures with isolated double bonds. Similarly, the structure with a naphthalene ring (Fig. 2(c’)) is more stable (by 775 kJ) than the struc- ture with five isolated double bonds (Fig. 2(c)). Between the structures containing two benzene rings, the structure 2(d’), where the two benzene rings are apart, is more stable (by 548 kJ) than where they form a biphenyl unit (Fig. 2(d)). On the basis of band structure calculations, Robertson [2] has proposed that sp2 carbons in DLCs tend to cluster together to form single aromatic rings or very small aromatic clus- ters. Structures (b)-(d) with a sufficient proportion of sp2 carbons could be models for these amorphous carbons.

In Fig. 3 we have plotted the values of the average coordination number, m, obtained from Eq. (1) for the various Cs4H, structures. We have also shown the theoretical line from Eq. (2) in the figure. We see that the m values from Eq. (1) for graphite-like structures shown in Fig. 1 (represented by circles in Fig. 3) vary linearly with xu, but the slope is smaller than that of the theoretical line predicted by Eq. (2). In the structures considered here, XH is increased by replacing one 3-coordinate carbon atom (sp2 carbon)

A

0

\

A

0 A 0

A

““t-----A 0.30 0.40 0.50 0.60

Atom fraction of Hydrogen

Fig. 3. A plot of the average coordination number, m, against the hydrogen atom fraction, xu, for diamond-graphite hybrids: circles, Cs4Hx structures of Fig. 1; triangles, Cs4H, structures of Fig. 2: squares, Cs4H, structures obtained by scheme (b) in Fig. 4. The straight line is from Eq. (2).

by one 4-coordinate carbon (sp3 carbon) and one l-coordinate hydrogen. While this decreases m with increase in XH, the decrease is not as much as is expected from Eq. (2). The m values from Eq. (l), for these structures, fall below the theoretical line up to XH = 0.35 and above the theoretical line when xu > 0.35. The composition with xu = 0.35 with about 30% sp3 carbons is close to that of amorphous carbon, whose structure could indeed be similar to that of Cs4Hd6 (Fig. l(d’)).

We have also plotted the m values from Eq. (1) against XH for various diamond-like structures. The m values of the structures shown in Fig. 2 (represented by triangles in Fig. 3) are all higher than the values from Eq. (2). In the structures of Fig. 2, sp2 carbons are created from sp3 carbons by scheme (a) of Fig. 4, wherein the creation of sp2 carbons is accompanied by the formation of C-H bonds. The sharp decrease in m with increase in xu happens because we are replacing one 4-coordinate atom with one 2-coordinate atom and one l-coordinate atom.

We have considered other ways of generating sp’ carbons from sp’ carbons, as shown in schemes (b) and (c) of Fig. 4. Scheme (b) involves a decrease in XH with increase in the number of sp2 carbons. The m values of such structures (shown by squares in

R. Sen et al./Journal of Molecular Structure 436-437 (1997) IILl

CH CH 8.00 r

( w

(a

H H

C ?-G-

C C

H C

C s- C C

C x C

C C

CH f (SP3)

C

#

C

C C

+2H

T

Fig. 4. Possible schemes for the transformation of sp’ carbons to sp’ carbons.

Fig. 3) fall well above the line from Eq. (2), but the slope is about the same. Thus, if we start with an sp’ network and introduce hydrogens by breaking the C-C bonds, so that the initial m value falls somewhere near the theoretical, and then introduce Sp2 carbons by scheme (b), we may obtain m values close to the line predicted by Eq. (2). This is not possible with 84 carbon atoms because the connectivity of the network breaks down at high hydrogen content. Scheme (c) in Fig. 4 does not allow a change in xn and is not relevant. Clearly, Eq. (2) fails in diamond-like structures with high proportions of sp3 carbons, unless some of the C-C bonds are broken to introduce hydrogens. Eq. (2) appears to be more applicable to carbon networks with high proportions of sp2 carbons.

3.2. Band gaps in diamond-graphite hybrids

Initial calculations on several hydrocarbon structures showed that the aromatic hydrocarbon

A

*

0.00 0.00 0.20 0.10 0.60 0.60 1.00

Fig. 5. Plot of the HOMO-LUMO gap, A, in Cl ,,)H I compositions (circles) against the fraction of sp’ carbons, Asp’). Triangles represent data on C120H, compositions. Experimental values are shown by stars.

C ,tOH3,, containing only sp2 carbons has a gap of 0.0009 eV and provides a good model for a graphitic sheet. We have progressively increased the number of sp3 carbons in this network, taking into account the different possible structures of such hybrids. As we progressively increase the number of sp’ carbons, we finally obtain the structure with only sp” carbons and with a high value of the gap (-9 eV). In Fig. 5 we have plotted the HOMO-LUMO gap, A, against the fraction of sp3 carbons in the CIIoH., system. (Note that the hydrogen content will also increase with the increase in the fraction of sp” carbons.) We see from Fig. 5 that the gap is in the 0.0-0.8 eV range up to an sp3 fraction of -0.4 and increases significantly when the sp3 fraction is increased further. Structures with a high sp’ fraction have gaps in the range 1 .O-4.0 eV. When the sp’ fraction is less than 0.4, we find a range of gaps (0.0-0.6 eV) for different structures with the same fraction of the sp” carbons because of the different ways of locating the sp’ carbons in the sp’ network. Such differences give rise to changes in the gap, but the changes are relatively small. When the sp’ carbons cluster together to form cyclohexane, decalin or the saturated forms of pyrene and coronene in the sp’ layer, the gap becomes very small

16 R. Sen et al/Journal of Molecular Structure 436-437 (1997) II-18

(--10e3 eV). Structures with a high sp” fraction (0.7-0.8) but low band gap (0.8 eV) are those where the sp* carbons form strips of fused benzene rings. Significant changes in the gap come about when we increase the number of sp3 carbons not merely at the edges but over the entire network.

The HOMO-LUMO gaps, A, in the hybrids of the general formula C , ioH, are described satisfactorily by the expression

A (in eV) =0.04 exp(4.34f)

where f is the fraction of sp3 carbons. The broken curve in Fig. 5 is that predicted by the above expression. Experimental values of the optical gaps reported in the literature [8] also show an exponential increase with the fraction of sp3 carbons, although they tend to be somewhat higher than those predicted by the above expression.

The variation in the HOMO-LUMO gap in a diamond network with progressive increase in the number of sp2 carbons is also of interest. The parent diamond network, C120H92, has a gap of 7.22 eV. Although the experimental optical gap for diamond is 5.5 eV, theoretical values of 7-10 eV are con- sidered to be good estimates [ 171. We have calculated the gap of the C ,20Hx system by increasing the number of sp2 carbons in the form of double bonds or aromatic rings as in diamond-like carbon. Thus Ci2aH9s, which has a benzene ring in the sp3 network, has a gap of 2.81 eV. The gap in pure benzene is, however, calculated to be 4.5 eV. Similarly, the gaps found in the structures containing isolated naphthalene and anthracene units are 1.84 eV and 1.34 eV respectively, which are much lower than the gaps calculated for the pure hydrocarbons (2.8 eV and 1.8 eV respectively). The band gaps of diamond-like C120Hx structures with varying sp2 content are shown in the A vs flsp3) plot in Fig. 5 (represented by triangles) to demonstrate that they are described reasonably well by the same expression, The experimental values of the gap, with their uncertainties, are also shown in Fig. 5 to illustrate that they are not far from the predicted trend.

The present study demonstrates how the HOMO- LUMO gap in diamond-graphite hybrids is a sensitive function of the sp”/sp* ratio as well as the structure of the hybrid. The marked variation in the gap when Asp”) is between 0.4 and 1 .O is noteworthy.

3.3. Fullerene-like structures with 7-membered rings

Electron microscopic investigation of the cathodic deposits obtained during arc vaporization of graphite has revealed bent nanotubes [lo] and that such bent nanotubes can occur due to the presence of heptagons (seven-membered rings).

Seven-membered rings give rise to negative curvature on a flat sheet of hexagons, whereas five- membered rings give rise to positive curvature. A heptagon/pentagon pair on opposite sides of a nanotube can thus cause the tube to bend. We have simulated such a bent tube with seven- and five-membered rings in Fig. 6a. In Fig. 6b we show a TEM image which may correspond to such bent

Fig. 6. (a) Structure of a bent nanotube containing seven- and five- membered rings: (b) TEM image of a nanotube with a negative curvature; (c) AFM image of a bent nanotubc.

R. Sen rt al./Journal of Moleculur Structure 436-437 (1997) I I-18 17

(b

Fig. 7. Small fullerene-like structures having varying numbers of pentagons and heptagons.

tubes. Fig. 6c gives an AFM image of a bent nanotube. Seven-membered rings can, in principle, occur in closed-cage structures which are fullerene-like.

In Fig. 7 we show a few of the small fullerene-like structures containing varying numbers of pentagons and heptagons, obtained by energy minimization. Fig. 7(a) represents C6” with twelve pentagons and zero heptagons. Fig. 7(b) shows Cg4 containing two heptagons and fourteen pentagons, this structure being highly symmetrical with the two heptagons present in the opposite ends. This would be one of the simplest of such fullerene-like molecules with 7-membered rings that chemists may one day synthesize. Fig. 7(c) shows C rZO with one heptagon and thirteen pentagons. Although the numbers of heptagons and pentagons are small in this case, the odd numbers make the structure unsymmetrical, requiring a larger number of hexagons to close the structure. Fig. 7(d) is that of CIA4 with four heptagons and sixteen pentagons. In the structures shown in Figs. 7(b)-(d), the heptagons and pentagons are isolated, but structures can also be built where pentagon and heptagon occur in pairs. A structure of CIo2 with a pentagon-heptagon pair is shown in Fig. 7(e), while a structure of Cg8 with a

Fig. 8. Fullerene-like and tube-like structures with S- and 7-membered rings showing unusual shapes.

pentagon-heptagon-pentagon triplet is given in Fig. 7(f). We can have unusual shapes with various combinations of pentagons, hexagons and heptagons. In Fig. 8 we show fullerene-like or tube-like structures with unusual shapes.

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