Scientific paper
Hyper-diamonds and Dodecahedral Architectures by Tetrapodal Carbon Nanotube Junctions
Katalin Nagy, Csaba L. Nagy, Erika Tasnadi, Gabriel Katona
and Mircea V. Diudea*
Babes-Bolyai University, Faculty of Chemistry and Chemical Engineering Cluj-Napoca,
Arany J. 11, 400028
* Corresponding author: E-mail: diudea@chem.ubbcluj.ro
Received: 29-06-2012
Abstract
Tetrapodal junctions are used to construct diamond-like networks and dodecahedral architectures. They can be associated with the already synthesized spongy carbon, consisting only of sp2 covalent carbon atoms, and the zeolites, periodic structures in the Euclidean space. In this paper, the structure and stability of two zigzag tetrapodal junctions are discussed. Series of objects are built up by connecting a various number of junctions. Geometry optimization and single point computations (total energy Etot and HOMO-LUMO gap energy Egap) were performed at the Hartree-Fock level of theory in view of evaluating their stability. The genus of such nanostructures was calculated from the number of consisting tetrapodal junctions.
Keywords: Tetrapodal junctions, hyper - structures, multitori
1. Introduction
In 1991 Mackay and Terrones1 studied the effect of introducing rings larger than the hexagon in the planar graphitic sheet. They designed structures embedded in surfaces of negative curvature, called ,,schwarzites", in honor of the German mathematician Schwarz.23 Modeling such structures led to the conclusion that they are more stable than the closed fullerenes and could compete in this respect with C60, the reference structure in Nanoscien-ce. Their stability has the origin in the very low strain induced by heptagons and octagons contained in such structures while keeping the sp2 character of the graphite. The genus of schwarzite4-6 units is 2 (for tetrahedral symmetry) or 3 (for octahedral symmetry). Such structures can be viewed as carbon nanotube junctions.
The tetrahedral/tetrapodal junctions are particularly interesting due to their similarity with the tetrahedral sp3 hybridized carbon atom: the valences are now nanotubes while the atom is an opened cage embedded in a surface of genus 2. Similar to the tetrahedrally coordinated carbon atom, a tetrapodal junction can be used to build various hyper-nanostructures such as dendrimers and multitori, which could act as molecular sieves.7
Multitori8 are complex structures consisting of more than one single torus;9 they are supposed to result by self-assembly of some repeating units/monomers, formed by opening of cages/fullerenes and appear in spongy carbon and in natural zeolites. Networks of sp2 spongy carbon, with large porosity, have already been synthesized10 and their properties studied.11,12 The high porosity of these materials has found applications in catalysis,13-15 in hydrogen storage,16 thermal insulators17,18 and molecular sieving,19 as well.
2. Results and Discussion
We have investigated the structure and stability of two tetrapodal junctions Tj40 and Tj52, their optimized geometries are presented in Figure 1. At each opening a (3,0) zigzag carbon nanotube can be attached, which were found to be narrow-gap semiconductors.20 The junctions were designed from the corresponding tetrahedral non IPR fullerenes21 C44 (T) and C56 (Td) by deleting the central atom in the pentagon triples. The tetrapodal junctions have only hexagonal rings, and can be connected by three octagons at each junction.22
Figure 1. Tetrapodal open units: Tj40 (left) and Tj52 (right)
2. 1. Energetics of Tetrapodal Junctions
The geometry optimization was performed (by Gaussian 09)23 at the Hartree-Fock level of theory, with HF/6-31G(d,p) set of parameters for the junctions and ful-lerenes and HF/3-21G* for the hyper pentagons and hexagons. All the structures were optimized in hydrogenated form. According to the simple Huckel model the studied fullerenes have pseudo-closed, while the junctions have pseudo-open electron configuration.24 Therefore, to obtain the highest possible symmetry in case of the Tj40-4 junction, charges were added as required. The framework of both junctions consists of four phenalene motifs, in case of Tj40 each fragment shares one bond with the neighboring motifs, and in case of Tj52 the fragments are disjoint separated be one bond. It has been reported25 that the 13-carbon phenalene cation an anion motif has a diatropic perimeter ring current like in the case of coronene,26 predicted by the ipsocentric model.
Data obtained using HF/6-31G(d,p) method are listed in Table 1 while the data computed by HF/3-21G* are listed in Table 2. The strain energy, evaluated according to POAV theory of Haddon27,28 is also included in these tables. One can see that the closed fullerenes (Table 1, entries 1, 4 and 6) show a higher strain in comparison to the open structures, the lowest value being for the IPR fullerene C60, the reference structure in nanoscience. Observe the higher symmetry of the charged T^"4 resulted in higher strain, when compared with the neutral lower symmetry Tj40. Even the presence of hydrogen atoms on the open structures prohibits a direct comparison with the close ful-lerenes, data in Table 1 suggest the stability of the tetrapo-dal structure stability is close to that of C60.
The strain decreases in structures with more than one repeating unit (see Table 2) while the HOMO-LUMO gap increases for the (hyper) cycles (Table 2, entries 3 and
6), with Tj40-hexagon being the most stable structure in Table 2. j40
2. 2. Hyper-Structures
The Tj40-hexagon, obtained by connecting six tetrapodal units and having the structure of the chair conformer of cyclohexane, is favored by the "intercalate" struc-
Tj40-units, one can
ture of Tj40-dimer.4 Further addition Tj build adamantane-like Tj40_ada and diamantane-like Tj40_dia and finally a hyper-diamond structure (Figure 2).
The Tj52-pentagon is favored by the "eclipsed" T52-dimer. Twelve such hyper-pentagons can self-assembly to form a hyper-dodecahedron (Figure 3).
Figure 2. Hyper-diamond substructures designed by Tj40_unit: Tj40_ada (left) and Tj40_dia (right)
Figure 3. Multitori by Tj52_unit: 10 units (left) and 20 units, closing a hyper-dodecahedron (right)
Design of the above structures was made by our software programs CVNET29 and Nano Studio.30
Table 1. Energetic data of tetrapodal junctions and their spanned fullerenes using HF/6-31G(d,p).
Table 2. Energetic data of tetrapodal junctions using HF/3-21G*.
Structures
Etot
(eV) / C atom
E
gap (eV)
Strain Energy (kcal/mol)
Sym
Struc-
E
tures (eV)/ C atom
E
gap (eV)
Strain Energy (kcal/mol)
Sym
1	C 44	-37.852	6.595	12.73	T	1	Tj40-dimer-37.768 Tj40-trimer-37.755	1.698	5.31	C3
2	T -■-140 T -4 Tj40	-38.021	6.681	5.80	D2	2		0.927	5.03	C2
3		-38.006	7.414	6.70	T	3	Tj40-hexagon-37.728	7.382	4.57	D3
4	C56	-37.854	5.505	10.65	Td	4	Tj52-dimer-37.741 Tj52-trimer-37.731 Tj52-pentagon-37.711	1.247	4.90	D3h
5	Tj52	-37.986	6.144	5.44	D2d	5		0.891	4.67	C,„
6	C 60	-37.864	7.418	8.256	Ih	6		6.755	4.25	D5h
2. 3. Genus in Multitori
An embedding is a representation of a graph on a surface S such that no edge-crossings occur.31 A polyhedral lattice, embedded in an orientable surface S obeys the Euler formula:32 v - e + f = x(M) = 2(1 - g), with x being the Euler characteristic. The genus g (i.e., the number of simple tori consisting a given network) is related to the Gaussian curvature of the surface S by means of Euler's characteristic x of S (Gauss-Bonnet theorem33) as: for g = 0 (case of sphere) x > 0 (positive curvature); for g = 1 (case of torus) x = 0 while for g > 1 (surfaces of high genera), X < 0, S showing a negative curvature. A surface is orientable, when it has two sides, or it is non-orientable, when it has only one side, like the Möbius strip.4,5
In open multitori built up from u tetrapodal junction units, the genus of structure is calculated as: g = u + 1, irrespective of the unit tessellation. In closed unit multitori, the genus can be calculated by formula g = u - Ec(uc x c/2) + 1 where c represents the number of closures per unit, while uc is the number of units with c closures. Table 3 lists examples representing the proof of the above theorem (in agreement with the Euler's formula) in both open and closed structures built up by using the tetrapodal junctions.
3. Conclusions
Tetrapodal junctions are proved to be stable structures possible to self-assembly in building diamond-like networks and hyper-dodecahedral architectures. They can be associated to the already synthesized spongy carbon, consisting only of sp2 covalent carbon atoms. Structural relatedness to zeolites was evidenced by the presence of large hollows.
Series of objects were built up by connecting a various number of junctions. The structure stability of tetrapodal junctions, performed at the Hartree-Fock level of theory was proved to be close to the reference fullerene C60.
Genus in both open and closed hyper-structures, consisting of tetrapodal units, was calculated both by applying the Euler's formula and by the newly proposed formula using the number of repeating units.
4. Acknowledgments
Csaba L. Nagy acknowledges the financial support of the Sectoral Operational Programme for Human Resources Development 2007-2013, co-financed by the European Social Fund, under the project number POSDRU 89/1.5/S/60189 with the title „Postdoctoral Programs for Sustainable Development in a Knowledge Based Society".
Tasnadi Erika acknowledges the financial support provided from programs co-financed by the Sectoral Operational Programme Human Resources Development, Contract POSDRU 6/1.5/S/3 - „Doctoral studies: through science towards society".
The work was also supported by the CNCSIS project PN-II ID_PCCE_129/2008.
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Table 3. Genus calculation in multi-tori by Euler formula and by the number of repeating units
	v	e	f6	fs	f	v-e+f	g	u
Tj40_ada T,40-dia	400 560	576 810	120 168	36 54	156 222	-20 -28	11 15	10 14
	v	e	f6	fs	f	v-e+f	g	u
Tj52- hyper-	1040	1530	360	90	450	-40	21	20
dodecahedron								
	v	e	f	fs	f	f	v-e+f	g
Tj40-ada-closed Tj40-dia-closed	416 580	624 870	120 168	36 54	48 60	204 282	-4 -8	3 5
	v	e	f	fs	fs	f	v-e+f	g
t,52 - hyper-dodecahedron-closed	1060	1590	360	90	60	510	-20	11
								
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Povzetek
Terapodalna stičišča smo uporabili pri konstrukciji diamantnih in do dekahhedralnih omrežij. Ta so lahko povezana z že sintetiziranim gobastim ogljikom, ki ima samo sp2 kovalentne ogljikove atome ali pa z zeoliti, to je, periodičnimi strukturami v Evklidskem prostoru. V tem članku obravnavamo strukturo in stabilnost dveh cik-cak tetrapodalnih stičišč. Izgradili smo vrsto objektov s pomočjo različnih povezav stičišč. Izvedli smo vrsto računov geometrijske in eno točkovne optimizacije (skupna energija Etot in HOMO-LUMO vrzelv energiji Egap) naravni Hartree-Fockteorije, da smo lahko ocenili stabilnost obravnavanih nanostruktur. Rodovno funkcijo tehnano struktur pa smo izračunali iz števila tetrapodalnih stičišč.