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.

Fig. 1
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Diamond D6: adamantane (left), diamantane (middle) and Fd3m network (right)

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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].

Fig. 2
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Lonsdaleite L6_: monomer (left), dimer (middle) and P63/mmc network (right) hyperdiamonds are covalently bonded fullerenes in crystalline forms, more or less related to the diamond D6 [17]. Several diamond-like networks have also been proposed [2, 18–20]. In a previous study, Diudea and Ilić [21] described some multi-tori (Fig. 3) constructed by using a unit designed by the map operation [1, 2, 22–24] sequence Trs(P 4(T)). These structures consist of all pentagonal faces, observing the triangles disappear (as faces) in the building process

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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).

Fig. 3
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Multi-tori M17 (left) and M57 (right) v = 972; e = 1770; f 5 = 684

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Fig. 4
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The centrohexaquinane C17 (left) and the multi-cage C57: v = 57; e = 94; r 5 = 42 (right)

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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).

Fig. 5
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SD5 (C57 based) triple periodic network: top view (left) and corner view (right)

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Table 1 Topology of the spongy SD5, a C57-based net
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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].

Fig. 6
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Dimerization of C17 (left) to C34 (right)—the repeating unit of dense diamond D5

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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.

Fig. 7
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Diamond D5_20_860 net (left) and D5_28_1022 co-net (right)

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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.

Fig. 8
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Intermediate structures originating in C34 unit: C51 (left) and 3 × C51 (right)

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Fig. 9
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Adamantane-like structures: ada_20_158 (left) and ada_28_213 (right)

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Fig. 10
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Diamantane-like structures: dia_20_226 net (left) and dia_28_292 co-net (right)

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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.

Fig. 11
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Lonsdaleite: L5_28_250 (side view, left), and its consisting hyper-hexagon L5_28_134 (side and top view, central, and right, respectively)

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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].

Table 2 B3LYP/6-31G(d,p) single point energy results
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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.

Table 3 Omega polynomial in Diamond D5_20 net function of k = 1, 2, ··· no. of ada_20 units along the edge of a (k,k,k) cuboid
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Table 4 Omega polynomial in D5_28 co-net function of k = no. ada_20 units along the edge of a (k,k,k) cuboid
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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.