Abstract
Diamond D5 is a hyperdiamond structured as coalesced C20 and C28 small fullerenes, in the ratio 2:1, and having up to 90 % pentagonal rings while the others being hexagonal. Design of several precursors, intermediates, and crystal networks was performed using our original software programs CVNet and Nano-Studio. Energetic data, calculated at DFT and DFTB levels of theory revealed a stability of these structures close to that of classical diamond. A lonsdaleite-like structure is also discussed. The topology of these networks is described in terms of the net parameters and the net characteristics in crystallographic terms.
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Introduction
Diamond D6, the beautiful classical diamond, with all-hexagonal rings of sp 3 carbon atoms (Fig. 1) crystallized in a face-centered cubic network (space group Fd3m), has kept its leading interest among the carbon allotropes, even in the nano-era, the period starting in 1985 with the discovery of C60 fullerene [1–3]. Its mechanical characteristics are of great importance, as some composites can overpass the resistance and stiffness of steel or other metal alloys. Synthetic diamonds can be produced by a variety of methods, including high pressure-high temperature HPHT static or detonation procedures [4–9], chemical vapor deposition CVD [10], ultrasound cavitation [11], or mechano-synthesis [12], under electronic microscopy.
Among the polymorphic diamondoids, structures having a significant amount of sp 3 carbon atoms [13], lonsdaleite [14] is the closest diamond relative [15], with its hexagonal network (space group P63/mmc, Fig. 2); it was discovered in a meteorite in the Canyon Diablo, Arizona, in 1967 and next synthesized. The evidence of diamond transformation into lonsdaleite was also reported [16].
Multi-tori MT are structures of high genera [1–3], consisting of more than one tubular ring. They are supposed to result by self-assembly of some repeating units (i.e., monomers) which can be designed by opening of cages/fullerenes or by appropriate map/net operations. Multi-tori appear in processes of self-assembling of some rigid monomers [25]. Zeolites [26] and spongy carbon [27, 28], also contain multi-tori. Multi-tori can be designed starting from the Platonic solids, by using appropriate map operations.
Diamond D5 network
Diamond D5, is the name proposed by Diudea [29] for diamondoids consisting mostly of pentagonal rings. Predominance of pentagons over the hexagonal rings is encountered in the spongy diamond SD5, a periodic structure constructed on the ground of the reduced graphs of multi-torus M57 (Fig. 3, right), namely C57 (Fig. 4, right),consisting of four C20 fullerenes any two such cages sharing a face and all sharing a central point. The central part of C57 is the centrohexaquinane skeleton C17 (Fig. 4, left), which is the reduced graph of multi-torus M17 (Fig. 3, left).
The nodes of diamond SD5 network (Fig. 5) consist of C57 units and the network is a triple periodic one. The number of atoms, bonds, and C57 monomers m, as well as the number of pentagons R [5] and the content in sp 3 carbon C (deg 4), given as a function of k—the number of monomers along the edge of a (k,k,k) cuboid. At limit, in an infinite net, the content of sp 3 carbon approaches 77 % (Table 1).
Crystallography of SD5 is as follows: the net is a 7-nodal net, with cubic symmetry Fm-3m, its point symbol being {5^3}16{5^5.8}36{5^6}17 3,3,4,4,4,4,4-c, with the stoichiometry (3-c)4(3-c)12(4-c)4(4-c)12(4-c)24(4-c)12(4-c).
C17 (Fig. 4, left) is supposed to dimerize, as centrohexaquinacene, by a synchron cycloaddition, to 2 × C17 = C34 (Fig. 6, right), which is the repeating unit of dense diamond D5 [30].
This dominant pentagon-ring diamond (Fig. 7, see also Table 3) is a mtn triple periodic, 3-nodal net, namely ZSM-39, type II clathrate, of point symbol net: {5^5.6}12{5^6}5 and 2[512] + [512 × 64] tiling and belongs to the space group: Fd3m. It is also known as fcc C34 structure, because of its face-centered cubic lattice.
In words, the network of D5 consists of C20 (i.e., dodecahedron, a 12-hedron) and C28 (i.e., hexakaidecahedron, a 16-hedron) fullerene cages, in ratio 2:1; each edge shares three cages and each carbon atom shares four cages. Looking from C28 fullerene, the co-net is like that in Fig. 7, right. The name of structures, hereafter, includes the type of net, number of cage atoms and the total number of atoms/points in the net.
Recall that clathrates represent foamy structures, like crystals of ice or some metal alloys [30–39]. They have been introduced in the late 40th to describe crystal structures with organic inclusions or chlorine hydrate structures or also pure carbon clathrate structures [32, 35–37]. An infinite fullerenic clathrate lattice is a polyhedral filling of the 3D space resulting by the coalescence of fullerenic cages, so that all atoms are of fourfold connected, with eclipsed sp 3 bonding, to their neighbors, and form exclusively five- or six-membered rings [35].
In a chemist view, the building of D5 network may start with the seed C17 structure and continue with modeling of some intermediates, the adamantane- and diamantane-like ones included (Figs. 8, 9, and 10). The structure ada_20_158 (Fig. 9, left), corresponds to adamantane (Fig. 1, left) in the classical diamond D6. The ada-like structure, starting from C28 can be seen in Fig. 9, right). Diamantane-like units can also be modeled, as in Fig. 10 (see for comparison the diamantane, Fig. 1, middle). In fact, there is one and the same triple periodic D5 network, built up basically from C20 and having as hollows the fullerene C28. The co-net D5_28 cannot be derived from C28 alone since the hollows of such a net consist of C57 units (a C20-based structure, see above) or higher tetrahedral arrays of C20 thus needing extra C atoms per ada-unit.
It is worthy to note the stabilizing effect of the wings in case of C34 (Fig. 6, right) in comparison to C20. Remark the efforts made by a series of bright scientists [40–44] to reach the dodecahedral cage C20, either as fullerene or hydrogenated species. Also remark the endeavor to synthesize the centrohexaquinane C17, both as oxygen-containing heterocycle [45, 46] or all carbon structure [47, 48].
Thus, the hyperdiamond D5_20/28 mainly consists of sp 3 carbon atoms building adatype repeating units (including C28 as hollows). The ratio C-sp 3/C-total trends to 1 in a large enough network. As the content of pentagons R [5] per total rings trend to 90 %, this, network was named the diamond D5.
Lonsdaleite L5 network
A lonsdaleite-like net was also proposed (Fig. 11). It consist of hyper-hexagons L5_28_134 (Fig. 11, central and right), of which nodes represent the C28 fullerene, was used as the monomer (in the chair conformation). The corresponding L5_20 co-net was also designed. The lonsdaleite L5_28/20 is partially superimposed on D5_20/28 net. In crystallographic terms, lonsdaleite L5 represents the mgz-x-d net, with the point symbol: {5^5.6}12{5^5.6}5; 4,4,4,4,4,4,4-c, and is a 7-nodal net.
Design of several hypothetical crystal networks was performed by using our original software programs CVNet [49] and Nano Studio [50]. Topological data were provided by the Nano Studio program.
Theoretical calculations
Energetic data, calculated at the hybrid B3LYP density functional level of theory using the standard polarized 6-31G(d,p) basis set show a good stability of the start and intermediate structures (single point energy results are summarized in Table 2). Geometry optimizations with and without symmetry constraints resulted in identical structures. All the calculations were performed using the Gaussian 09 package [51].
Partially hydrogenated molecular fragments of the hyperdiamond and hyperlonsdaleite networks have been evaluated for stability, data proving a pertinent stability of the D5 diamond. The energy of the precursors ranges between that of the adamantine C10H16–T d and the C60–I h fullerene, taken as reference structures (see Table 2).
Due to the huge number of atomic components, the large carbon networks could not be properly described in the framework of the ab initio or the conventional DFT molecular theories. In contrast, the density functional-based tight-binding method combined with the selfconsistent charge technique (SCC–DFTB) [52] can be considered as an adequate solution for treating large biologically interested or nanoscaled molecular materials with nearly good accuracy as obtained in the case of high-level theoretical methods [53–55].
The carbon network structure of Diamond D5_20_860 net and D5_28_1022 co-net (see Fig. 7) were optimized using the DFTB + program [56, 57] considering the numerical conjugated gradient method. Hydrogen atoms were added to the external carbon atoms of the network structures, in order to keep the charge neutrality and the sp 3 character of the C–C bonds at the network surface. In both cases, energetically stable geometry structures were obtained, where the repeating unit of dense diamond D5 is kept.
Identification of the equivalent carbon atoms in the neighboring units of the 3 × 3 × 3 super-cell along the main symmetry axes, envisaged a well-defined triclinic lattice, with the following parameters: a = b = c = 6.79 Å, as well as, α = 60º, β = 120º, γ = 120º, even the most symmetrical structure is fcc-one [30]. Density of the D5 network was calculated to be around 2.8 g/cm3.
Analyzing the C–C bond distances in these carbon networks, these values vary in a very narrow distance domain of 1.50–1.58 Å, showing that all C–C bonds present sp 3 valence orbital hybridization. Considering the one-electron energy levels of the HOMO and LUMO, a very large energy gap can be observed for both D5_20_860 net (E HOMO = −5.96 eV, E LUMO = +2.10 eV, ΔE HOMO−LUMO = 8.06 eV) and D5_28_1022 co-net (E HOMO = −6.06 eV, E LUMO = +2.45 eV, ΔE HOMO−LUMO = 8.51 eV) structures, which indicates an insulating behavior for this carbon network, similar to the classical diamond D6 [35, 39, 58].
Topology of diamond D5
Topology of diamond D5, in a (k,k,k) cuboid (see Fig. 7), is presented in Table 3 and Table 4: formulas to calculate the number of atoms, number of rings, and the limits (at infinity) for the ratio of sp 3 C atoms over the total number of atoms and also the ratio R [5] over the total number of rings are given function of k. The network parameters are: a = b = c = 6.79 Å, while the angles between axes are 60, 120, and 120°, respectively.
Conclusions
Diamond D5 is a hyperdiamond structured on the type II clathrate network, also known as fcc-C34. Design of several precursors, intermediates, and crystal networks was performed using our original software programs CVNet and Nano-Studio developed at TOPO GROUP CLUJ. Energetic data, calculated at DFT and DFTB levels of theory, revealed a stability of these structures close to that of classical diamond and its units. A lonsdaleite-like network is also discussed. The topology of these structures is described in terms of the net parameters and the net characteristics in crystallographic terms.
References
Diudea MV (2010) Nanomolecules and nanostructures: polynomials and indices. Univ, Kragujevac
Diudea MV, Nagy CL (2007) Periodic nanostructures. Springer, Dordrecht
Nagy CL, Diudea MV (2005) In: Diudea MV (ed) Nanostructures, novel architecture. NOVA, New York
Aleksenskiǐ AE, Baǐdakova MV, Vul AY, Davydov VY, Pevtsova YA (1997) Diamondgraphite phase transition in ultradisperse-diamond clusters. Phys Solid State 39:1007–1015
Williams OA, Douhéret O, Daenen M, Haenen K, Osawa E, Takahashi M (2007) Enhanced diamond nucleation on monodispersed nanocrystalline diamond. Chem Phys Lett 445:255–258
Decarli PS, Jamieson JC (1961) Formation of diamond by explosive shock. Science 133:1821–1822
Osawa E (2008) Monodisperse single nanodiamond particulates. Pure Appl Chem 80:1365–1379
Osawa E (2007) Recent progress and perspectives in single-digit nanodiamond. Diam Relat Mater 16:2018–2022
Dubrovinskaia N, Dub S, Dubrovinsky L (2006) Superior wear resistance of aggregated diamond nanorods. Nano Lett 6:824–826
Lorenz HP (1995) Investigation of TiN as an interlayer for diamond deposition on steel. Diam Relat Mater 4:1088–1092
Khachatryan AK, Aloyan SG, May PW, Sargsyan R, Khachatryan VA, Baghdasaryan VS (2008) Graphite-to-diamond transformation induced by ultrasound cavitation. Diam Relat Mater 17:931–936
Tarasov D, Izotova E, Alisheva D, Akberova N, Freitas RA Jr (2011) Structural stability of clean, passivated, and partially dehydrogenated cuboid and octahedral nanodiamonds up to 2 nanometers in size. J Comput Theor Nanosci 8:147–167
Robertson J (2002) Diamond-like amorphous carbon. Mater Sci Eng R 37:129–281
Frondel C, Marvin UB (1967) Lonsdaleite, a hexagonal polymorph of diamond. Nature 214:587–589
Balaban AT, Ragé Schleyer PV (1978) Systematic classification and nomenclature of diamond hydrocarbons-I. Graph-theoretical enumeration of polymantanes. Tetrahedron 34:3599–3609
He H, Sekine T, Kobayashi T (2002) Direct transformation of cubic diamond to hexagonal diamond. Appl Phys Lett 81:610–612
Ivanovskii AL (2008) Hyperdiamonds. Russ J Inorg Chem 53:1274–1282
Sunada T (2008) Crystals that nature might miss creating. Notices Am Math Soc 55:208–215
Diudea MV, Bende A, Janežič D (2010) Omega polynomial in diamond-like networks. Fuller Nanotub Carbon Nanostruct 18:236–243
Hyde ST, O’Keeffe M, Proserpio DM (2008) A short history of an elusive yet ubiquitous structure in chemistry, materials, and mathematics. Angew Chem Int Ed 47:7996–8000
Diudea MV, Ilić A (2011) All-pentagonal face multi tori. J Comput Theor Nanosci 8:736–739
Diudea MV (2004) Covering forms in nanostructures. Forma (Tokyo) 19:131–163
Diudea MV, Stefu M, John PE, Graovacc A (2006) Generalized operations on maps. Croat Chem Acta 79:355–362
Diudea MV (2005) Nanoporous carbon allotropes by septupling map operations. J Chem Inf Model 45:1002–1009
Diudea MV, Petitjean M (2008) Symmetry in multi tori. Symmetry Cult Sci 19:285–305
Anurova NA, Blatov VA, Ilyushin GD, Proserpio DM (2010) Natural tilings for zeolite-type frameworks. J Phys Chem C 114:10160–10170
Benedek G, Vahedi-Tafreshi H, Barborini E, Piseri P, Milani P, Ducati C, Robertson J (2003) The structure of negatively curved spongy carbon. Diam Relat Mater 12:768–773
Barborini E, Piseri P, Milani P, Benedek G, Ducati C, Robertson J (2002) Negatively curved spongy carbon. Appl Phys Lett 81:3359–3361
Diudea MV (2010) Diamond D5, a novel allotrope of carbon. Studia Univ Babes-Bolyai Chemia 55:11–17
Benedek G, Colombo L (1996) Hollow diamonds from fullerenes. Mater Sci Forum 232:247–274
Delgado-Friedrichs O, O’Keeffe M (2006) On a simple tiling of Deza and Shtogrin. Acta Crystallogr A 62:228–229
Aste T, Weaire D (2008) The pursuit of perfect packing. Taylor and Francis, Bristol
Delgado-Friedrichs O, O’Keeffe M (2010) Simple tilings by polyhedra with five- and sixsided faces. Acta Crystallogr A 66:637–639
Sikirić MD, Delgado-Friedrichs O, Deza M (2010) Space fullerenes: a computer search for new Frank–Kasper structures. Acta Crystallogr A 66:602–615
Blase X, Benedek G, Bernasconi M (2010) In: Colombo L, Fasolino A (eds) Computer-based modeling of novel carbon systems and their properties beyond nanotubes. Springer, New York
Pauling L, Marsh RE (1952) The structure of chlorine hydrate. Proc Natl Acad Sci USA 38:112–118
Powell HM (1948) The structure of molecular compounds. Part IV. Clathrate compounds. J Chem Soc 1:61–72
Delgado-Friedrichs O, O’Keeffe M, Yaghi OM (2006) Three-periodic nets and tilings: edge-transitive binodal structures. Acta Crystallogr A 62:350–355
Adams GB, Okeeffe M, Demkov AA, Sankey OF, Huang YM (1994) Wide-band-gap Si in open fourfold-coordinated clathrate structures. Phys Rev B 49:8048–8053
Prinzbach H, Wahl F, Weiler A, Landenberger P, Wörth J, Scott LT, Gelmont M, Olevano D, Sommer F, Von Issendorff B (2006) C20 carbon clusters: fullerene-boat-sheet generation, mass selection, photoelectron characterization. Chem Eur J 12:6268–6280
Paquette LA, Balogh DW, Usha R, Kountz D, Christoph GG (1981) Crystal and molecular structure of a pentagonal dodecahedrane. Science 211:575–576
Prinzbach H, Weller A, Landenberger P, Wahl F, Wörth J, Scott LT, Gelmont M, Olevano D, Issendorff BV (2000) Gas-phase production and photoelectron spectroscopy of the smallest fullerene, C20. Nature 407:60–63
Saito M, Miyamoto Y (2001) Theoretical identification of the smallest fullerene, C20. Phys Rev Lett 87:355031–355034
Eaton PE (1979) Towards dodecahedrane. Tetrahedron 35:2189–2223
Simmons Iii HE, Maggio JE (1981) Synthesis of the first topologically non-planar molecule. Tetrahedron Lett 22:287–290
Paquette LA, Vazeux M (1981) Threefold transannular epoxide cyclization. Synthesis of a heterocyclic C17-hexaquinane. Tetrahedron Lett 22:291–294
Gestmann D, Pritzkow H, Kuck D (1996) Partially benzoanellated centrohexaquinanes: oxidative degradation of centropolyindanes by using ruthenium(VIII) oxide and ozone. Liebigs Ann 9:1349–1359
Kuck D (2006) Three-dimensional hydrocarbon cores based on multiply fused cyclopentane and indane units: centropolyindanes. Chem Rev 106:4885–4925
Stefu M, Diudea MV (2005) CageVersatile CVNet. Babes-Bolyai University, Cluj-Napoca
Nagy CL, Diudea MV (2010) Nano studio. Babes-Bolyai University, Cluj-Napoca
Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Scalmani G, Barone V, Mennucci B, Petersson GA, Nakatsuji H, Caricato M, Li X, Hratchian HP, Izmaylov AF, Bloino J, Zheng G, Sonnenberg JL, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao O, Nakai H, Vreven T, J. A. Montgomery J, Peralta JE, Ogliaro F, Bearpark M, Heyd JJ, Brothers E, Kudin KN, Staroverov VN, Kobayashi R, Normand J, Raghavachari K, Rendell A, Burant JC, Iyengar SS, Tomasi J, Cossi M, Rega N, Millam JM, Klene M, Knox JE, Cross JB, Bakken V, Adamo C, Jaramillo J, Gomperts R, Stratmann RE, Yazyev O, Austin AJ, Cammi R, Pomelli C, Ochterski JW, Martin RL, Morokuma K, Zakrzewski VG, Voth GA, Salvador P, Dannenberg JJ, Dapprich S, Daniels AD, Farkas Ö, Foresman JB, Ortiz JV, Cioslowski J, Fox DJ (2009) Gaussian 09, Revision A.02. Gaussian Inc., Wallingford
Elstner M, Porezag D, Jungnickel G, Elsner J, Haugk M, Frauenheim T, Suhai S, Seifert G (1998) Self-consistent-charge density-functional tight-binding method for simulations of complex materials properties. Phys Rev B 58:7260–7268
Elstner M, Jalkanen KJ, Knapp-Mohammady M, Frauenheim T, Suhai S (2000) DFT studies on helix formation in N-acetyl-(l-alanyl)(n)-N’-methylamide for n = 1−20. Chem Phys 256:15–27
Elstner M, Jalkanen KJ, Knapp-Mohammady M, Frauenheim T, Suhai S (2001) Energetics and structure of glycine and alanine based model peptides: approximate SCC–DFTB, AM1 and PM3 methods in comparison with DFT, HF and MP2 calculations. Chem Phys 263:203–219
Elstner M, Hobza P, Frauenheim T, Suhai S, Kaxiras E (2001) Hydrogen bonding and stacking interactions of nucleic acid base pairs: a density functional-theory based treatment. J Chem Phys 114:5149–5155
DFTB+ 1.1 is a DFTB implementation, which is free for non-commercial use. For details, see: http://www.dftbplus.info
Aradi B, Hourahine B, Frauenheim T (2007) DFTB+, a sparse matrix-based implementation of the DFTB method. J Phys Chem A 111:5678–5684
Nesper R, Vogel K, Blöchl PE (1993) Hypothetical carbon modifications derived from zeolite frameworks. Angew Chem Int Ed 32:701–703
Acknowledgments
Cs.L.N. 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”.
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Diudea, M.V., Nagy, C.L. & Bende, A. On diamond D5 . Struct Chem 23, 981–986 (2012). https://doi.org/10.1007/s11224-012-0040-0
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DOI: https://doi.org/10.1007/s11224-012-0040-0