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7.1 Introduction

In the nano-era, a period starting with the discovery of C60, in 1985, the carbon allotropes played a dominant role. Among the carbon structures, fullerenes (zero dimensional), nanotubes (one dimensional), graphene (two dimensional), diamond, and spongy carbon (three dimensional) were the most studied (Diudea 2005, 2010a; Diudea and Nagy 2007), both from theoretical reasons and applications perspective.

Diamond D6, the beautiful classical diamond, with all-hexagonal rings of sp3 carbon atoms (Fig. 7.1), crystallized in the face-centered cubic fcc network (space group Fd-3m), has kept its leading interest among the carbon allotropes, in spite of the “nano” varieties (Decarli and Jamieson 1961; Aleksenski\( \check{\i} \) et al. 1997; Osawa 2007, 2008; Williams et al. 2007; Dubrovinskaia et al. 2006). Its aesthetical appeal and mechanical characteristics are of great importance in jewelry and industry. Synthetic diamonds are currently produced by a variety of methods, including high pressure-high temperature (HPHT), chemical vapor deposition (CVD), and ultrasound cavitation (Khachatryan et al. 2008).

Fig. 7.1
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Diamond D6 (left) and its repeating unit, adamantane (right)

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However, the diamond D6 is not unique: a hexagonal network called lonsdaleite (space group P63/mmc) (Frondel and Marvin 1967) was discovered in a meteorite in the Canyon Diablo, Arizona, in 1967. Several diamond-like networks have also been proposed (Diudea et al. 2010; Hyde et al. 2008).

In a previous study, Diudea and Ilić (2011) described some multi-tori (i.e., structures showing multiple hollows (Diudea and Petitjean 2008)); one of them is illustrated in Fig. 7.2, left.

Fig. 7.2
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A multi-torus (left) and its reduced graph C17 (right), the seed of diamond D5

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The reduced graph of this multi-torus provided the structure for the seed of diamond D5: C17 (Fig. 7.2, right) consisting of a tetravalent atom surrounded by six pentagons, the maximum possible number of pentagons around an sp3 carbon atom. According to the chemical nomenclature, C17 is a centrohexaquinane, a class of structures previously studied by Gund and Gund (1981), Paquette and Vazeux (1981), and more recently by Kuck (1984, 2006) and Kuck et al. (1995).

Diamond D5 is the name given by Diudea to diamondoids consisting mostly of pentagonal rings (Diudea 2010a, b; Diudea and Ilić 2011). D5 is a hyperdiamond built up in the frame of the trinodal mtn structure, while its seed is eventually the centrohexaquinane C17. However, D5 belongs to the family of Clathrates with the point symbol net {5^5.6}12{5^6}5 and 2[512] + [512 × 64] tiling and belongs to the space group Fd3m (Delgado-Friedrichs et al. 2005). It is precisely type II clathrate, also called C34 (Blasé et al. 2010), of which Si34-analogue was already synthesized.

C17 can dimerize to 2 × C17 = C34 (Fig. 7.3), the repeating unit, in crystallographic terms, of the diamond D5 network. Thus, D5 (Fig. 7.4) and fcc-C34 are herein synonyms (Fig. 7.5).

Fig. 7.3
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A joint of two C17 units (left) to give a dimer C34 (right), the repeat unit (in crystallographic terms) of the diamond D5 network

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Fig. 7.4
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Diamond D5_20_860 net (left) and D5_28_1022 co-net (right)

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Fig. 7.5
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Diamond D6: adamantane (left), diamantane (right)

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In a chemist’s view, the building of D5 network may start with the seed C17 and continue with some intermediates, the adamantane- and diamantane-like ones included (Figs. 7.6, 7.7, and 7.8). The structure ada_20_158 (Fig. 7.7, left) corresponds to adamantane (Fig. 7.5, left) in the classical diamond D6. In crystallochemical terms, an adamantane-like structure, as ada_20_158, is the monomer which will probably condense to form the D5 network (Fig. 7.4). The ada-like structure, starting from C28 can be seen in Fig. 7.7, right. Diamantane-like units can also be modeled, as in Fig. 7.8 (compare with the diamantane, Fig. 7.5, right). 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. 7.3, right) in comparison to C20.

Fig. 7.6
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Intermediate structures originating in C34 unit: C51 (left) and 3 × C51 (right)

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Fig. 7.7
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Adamantane-like structures: ada_20_158 (left) and ada_28_213 (right)

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Fig. 7.8
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Diamantane-like structures: dia_20_226 net (left) and dia_28_292 co-net (right)

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Remark the efforts made by a series of bright scientists (Prinzbach et al. 2006; Paquette et al. 1981; Saito and Miyamoto 2001; Eaton 1979) 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 (Simmons and Maggio 1981; Paquette and Vazeux 1981) or all-carbon structure (Gestmann et al. 2006; Kuck 2006). Thus, the hyperdiamond D5_20/28 mainly consists of sp3 carbon atoms building ada-type repeating units (including C28 as hollows). The ratio C-sp3/C-total trends to one in a large enough network. As the content of pentagons R[5] (Aleksenski\( \check{\i} \) et al. 1997; Williams et al. 2007) per total rings trend to 90 %, this network was named the diamond D5 (Diudea 2010a, b).

In the above symbols, “20” refers to C20 and “28” refers to C28, while the last number counts the carbon atoms in structures.

7.2 Method

In a study of structural stability, performed by Szefler and Diudea (2012), ab initio calculations and molecular dynamics have been used. The four structures (Figs. 7.9 and 7.12) based on C17 skeleton, as all-carbon or partly oxygenated derivatives, were optimized at the Hartree-Fock (HF) (HF/6-31G**) and DFT (B3LYP/6-311+G**) levels of theory and submitted to molecular dynamics (MD) procedure. All calculations were performed in gas phase by Gaussian 09 (Gaussian 09 software package 2009) while MD calculations were done in vacuum, using Amber 10.0 software (Case et al. 2005). The single-point energy minima obtained for the investigated structures are shown in Table 7.1. Before MD, the atomic charges were calculated according to Merz-Kollmann scheme via the RESP (Wang et al. 2000) procedure, at HF/6-31G** level. The AMBER force field (Wang et al. 2004) was used for dynamic trajectory generation. There were several steps of molecular dynamics. After stabilization of energies and RMSD values during run, the actual molecular dynamics were performed, in a cascade way. Each tested system was heated by 20 ps while MD simulations were 100 ns long. The visualizations were prepared in the GaussView program. After MD run, the values of RMSD and energies of analyzed structures were recorded: total energy (E tot), kinetic energy (E kin), and potential energy (E pot). In the analysis, averaged values of all generated points of energies and values of RMSD in every 1 ps of MD were used.

Fig. 7.9
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C17_hexaquinane trioxo derivatives: Paquette P1 (left) and Diudea, D1 (middle) and D2 (right)

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Table 7.1 The single-point energies of the optimized structures at DFT (B3LYP/6-311+G**) level of theory
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The stability of 12 other substructures was investigated by Kyani and Diudea (2012) by performing a molecular dynamics (MD) computer simulation, using an empirical many-body adaptive intermolecular reactive empirical bond-order (AIREBO) potential energy function. All the diamond D5 substructures are fully hydrogenated ones. The studied structures were optimized at the semiempirical PM3 level of theory and then submitted to the MD simulation procedure. Canonical ensemble molecular dynamics was used for this simulation. Within this ensemble, the number of atoms N, the volume V, and the temperature T are considered constants while velocities are scaled with respect to T, ensuring that the total kinetic energy, and hence the temperature, is constant (isokinetic MD). The initial velocities follow the Maxwell distribution. The AIREBO potential energy function (PEF) developed for hydrocarbons (Stuart et al. 2000), as provided by LAMMPS software (Plimpton 1995), was used to investigate the stability of nanostructures at increasing temperatures. This parameterized potential adds Lennard-Jones and torsional contributions to the many-body REBO potential (Brenner 2000; Brenner et al. 2000). It is similar to a pairwise dispersion-repulsion potential, while adding a bond-order function modulates the dispersion term and incorporates the influence of the local atomic environment. Through this interaction, individual atoms are not constrained to remain attached to specific neighbors, or to maintain a particular hybridization state or coordination number. Thus, at every stage of the simulation, the forming and breaking of the bonds is possible. This potential is derived from ab initio calculations, and therefore, it is well adapted to classical molecular simulations of systems containing a large number of atoms such as carbon nanostructures. The equations of particle motion were solved using the Verlet algorithm (Verlet 1967, 1968), and the temperature was gradually increased by 100 K at each run. One time step was taken to be 10–16 s and at each run a relaxation with 5,000 time steps was performed. The root-mean-square deviation (RMSD) of the atoms was used as criterion for examining the stability of the simulated structures.

7.3 Results and Discussion

Stability evaluation was performed on four hypothetical seeds of D5, the all-carbon structure C17 (Fig. 7.2, right) and three trioxa derivatives of C17. The isomer in Fig. 7.9, left, was synthesized by Paquette and Vazeux (1981) and is hereafter denoted as P1. Other two structures, denoted as D1 and D2 (Fig. 7.9, middle and right), were proposed (Szefler and Diudea 2012), as possibly appearing in rearrangements of the Paquette’s P1 structure. The last two structures would be the appropriate ones in the next step of dimerization to C34, in fact the repeating unit of D5 (Blasé et al. 2010).

The stability of molecules was evaluated both in static and dynamic temperature conditions. The isomer D1 seems the most stable among all studied structures, as given by optimization in gas phase at DFT level (Table 7.1). In a decreasing order of stability, it follows P1 and D2. However, at MD treatment, the all-carbon C17 appears the most stable, even at DFT level is the last one. This is probably because the C–C bond is more stable at temperature variations (see Fig. 7.12, below).

In MD, C17 keeps its structure up to about 1,800 K, while its destruction starts at 2,000 K (Tables 7.2 and 7.4, Figs. 7.10 and 7.11). Kuck has reported a centrohexaindane as the most symmetric structure in this series but also a benzo-centrohexaquinane (Kuck et al. 1995; Kuck 2006) as the last step structure in the synthesis of a nonplanar 3D structure, designed according to mathematical rules. However, in the synthesis of centrohexaquinane derivatives, C17 remained yet elusive.

Table 7.2 The average total energy (E tot) values estimated, by MD, on geometries in the gas phase
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Fig. 7.10
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The plot of total energy (E tot) versus temperature (TEMP)

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Fig. 7.11
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The plot of RMSD versus temperature (TEMP)

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Very close to C17 behaves the oxygen-containing isomer D1, as expected from its highest stability at DFT level.

Despite, in molecular dynamics, a very long time (100ns) was leaded, it is believed that prolonged annealing at 1,800 K for both P1 and D1 isomers finally resulted in the destruction of these molecules. Thus, P1 and D1 isomers behave similarly in MD conditions. The isomer D2 was the least stable one, as the largest RMSD values were recorded for this isomer.

According to molecular dynamics, it is clear that increasing the temperature resulted in higher values of energy and RMSD of all the analyzed structures, with high values of correlation. The plots of E tot versus temperature for all tested systems are given in Fig. 7.10, while for RMSD, the plots are given in Fig. 7.11. As expected, the correlations in the RMSD plot are a little lower than those for E tot. The MD calculations, listed in Tables 7.2, 7.3, and 7.4, show the following.

Table 7.3 The values of standard deviations of E tot at a given temperature (see the center of each slide) for the four investigated structures
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Table 7.4 The averaged RMSD values estimated by molecular dynamics (MD) on the geometries in the gas phase
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As can be seen from Tables 7.2 and 7.3, the values of standard deviations of the averaged values of E tot are closely correlated with the values of temperature, in the range the molecular dynamics simulations were done. The values of these standard deviations at a given temperature are similar for all four studied structures, due to their structural relatedness. The smallest values of the RMS deviation are observed for C17, with the lowest values of standard deviation (δ) at all the studied values of temperature (Table 7.4).

In the case of P1, one can see a similar behavior but somewhat with larger values of RMSD, compared to the all-carbon structure C17 (Table 7.4). It confirms the structural stability of the above structures. The largest values of the RMS deviation were recorded for D2 isomer (Table 7.4 and Fig. 7.11). Visualization of the structural changes (first step destruction, the right column) is presented in Fig. 7.12.

Fig. 7.12
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The structure of the tested hypothetical seeds of the diamond D5 during molecular dynamics

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When discussing about diamonds, we consider structures consisting mostly of sp3 hybridized carbon atom. The molecular dynamic with formulation and parametrization intended for carbon system is REBO (Tersoff 1988a, b; Abell 1985). The Tersoff’s and Brenner’s (Brenner 1990, 1992) models could describe single-, double-, and triple-bond energies in carbon structures such as hydrocarbons and diamonds, where extended Tersoff’s potential function is extended to radical and conjugated hydrocarbon bonds by introducing two additional terms into the bond-order function. Compared to the classical first-principle and semiempirical approaches, the REBO model is less time-consuming. In recent years, the REBO model has been widely used in studies concerning mechanical and thermal properties of carbon nanotubes (Ruoff et al. 2003; Rafii-Tabar 2004).

Molecular dynamic MD calculations using the REBO model were performed by Kyani and Diudea (2012) on the structures listed in Figs. 7.13 and 7.14. The last number in the symbol of structures refers to the number of carbon atoms.

Fig. 7.13
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C20-based structures; 206_282_110 a substructure of D5_20/28 (Reproduced from Central European Journal of Chemistry 2012, 10(4), 1028–1033)

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The main reason for doing calculations on hydrogenated species, although the fragments can appear as non-hydrogenated ones, is the sp3 hybridization of carbon atoms in the diamond structures. Thus, the four-valence state is preserved.

PM3 calculations show the hexagonal hyper-rings more stable than the pentagonal ones, either with a hollow or filled one as in case of lens-like structures (Table 7.5, entries 5, 6 and 10). There is one exception; the empty hexagon of C28 fullerenes (entry 9) is less stable than the corresponding empty pentagon (entry 7). The hexagonal filled ring structures are more stable than the empty ones, except the empty hexagon 206_H60 (entry 5), made from C20, which seems to be the most stable, as isolated structure, herein discussed. Similarly, C20H20 (entry 1) is the stabilized form of the most reactive/unstable fullerene C20. The stabilizing effect of ring filling is a reminiscence of the infinite crystal lattice, whose substructures are 206_282H68 (diamond D5 net) and 286_202H92 (lonsdaleite L5 net).

Table 7.5 PM3 energies calculated on fully hydrogenated species
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Fig. 7.14
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C28-based structures; 286_20_134 a substructure of L5_28/20. The red lines show the core substructure, that is C20 the smallest fullerene (Reproduced from Central European Journal of Chemistry 2012, 10(4), 1028–1033)

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As in studies of the stability of hypothetical seeds of the diamond D5 by using Amber 10.0 for MD, also here we could see that with increasing temperature, the energy of the systems increases, what would be expected. The total energy versus temperature for three of the considered systems is illustrated in (Fig. 7.15).

Fig. 7.15
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Total energy versus temperature of some of the diamond D5 substructures: 205_202H50 (solid line, Table 7.5, entry 4), 206_282H68 (simple line, Table 7.5, entry 6), and 285_202H80 (dotted line, Table 7.5, entry 8) (Reproduced from Central European Journal of Chemistry 2012, 10(4), 1028–1033)

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MD simulations on increasing temperature evolution show that the studied structures are stable up to 2,000 K, for the C20-based structures (Fig. 7.13), and up to 1,500–2,000 K, for the C28-based structures (Fig. 7.14). The three numbers at the figure bottom represent temperatures, in K degree, for: geometry modifications, topology changes, and major destruction of the structure. Where there are only two data, the topological changes were not observed.

The structures in Fig. 7.16 show elongated bonds (i.e., broken bonds, marked by orange color) at temperatures above 2,000 K. Question about structure preserving must be addressed when more than one broken bond will appear (see the bottom row, Fig. 7.16). Above 2,500 K, the complete destruction is expected for all the three structures shown in Fig. 7.16; for two structures (those with only two temperatures on their bottom) the topological changes were not observed. In case of the basic fullerenes, the data are: C20H20, 2,700 and 3,000 K, and C28H28, 2,500, 2,600, and 3,000 K. Among the frequent topological changes, the most important is the expansion of two pentagons sharing an edge to octagon and also the apparition of trigons, squares, or larger rings before the structure dramatically decomposes. It is obvious that the structural changes will affect the energetics of the system.

Fig. 7.16
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Relaxed structures of 205_202_85, 206_282_85 and 285_202_85 at various temperatures. The orange colors mark the broken/formed bonds (Reproduced from Central European Journal of Chemistry 2012, 10(4), 1028–1033)

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The MD data agree with the PM3 data; in fact, the C20-based structures are more stable than the C28-ones, which corresponds to the higher stability of diamond D5 compared to lonsdaleite L5 (Aste and Weaire 2008). At higher temperature, the five-fold hyper-rings seem to be more stable, at least in the isolated fragments (see Figs. 8.13 and 8.14). Remind that such fragments can appear either in synthesis of D5 or in its destruction, their knowledge thus being of real interest. It is important to know the limit temperature, e.g., in the annealing process of repeating unit C34 in the possible synthesis of D5. Conversely, in the analysis of such diamondoids, the possible fragments appearing in the destruction of their lattice must be known.

7.4 Conclusions

In this chapter, structural stability of four seeds of the diamond D5 and several substructures/fragments related to the D5 diamond were investigated. In the first case, it was evaluated both in static and dynamic temperature conditions by molecular dynamics (MD). During MD, the all-carbon C17 appeared the most resistant to changes in temperature. Structural and energetic stability of the other three seeds of D5 vary both with the values of temperature and evolution time in molecular dynamics and the arrangement of oxygen atoms in the molecules. Among all the studied structures, the D2 isomer is the most sensitive to changes in temperature. After optimization by B3LYP, D1 isomer seemed to be the most stable one. The structure stability of D1 and P1 isomers in MD are similar. These two isomers are only slightly more sensitive to temperature as compared with the all-carbon C17.

Other substructures/fragments related to the D5 diamond (and its relative L5 lonsdaleite) were constructed and investigated for stability, by both static PM3 calculation and molecular dynamics simulation procedures as well. The results show a good stability of several hyper-rings made from the small fullerenes C20 and C28, modulated function of the central hollow, and the type consisting of small cages. At normal temperature, the hexagonal hyper-rings are more stable (as given by PM3 data), while at higher temperature, the pentagonal ones appear more stable, at least as isolated fragments. The substructures belonging to D5 and L5 showed a pertinent stability, possibly increased in the infinite corresponding lattice. The actual study employed the molecular dynamics simulation in finding the temperature limits for the most important events in a molecule: changes in topology and next its destruction.

These results could be useful in guiding further reactions, e.g., the dimerization to C34 and condensation to adamantane-like structures, finally leading to the diamond D5 or, in general, in the design and synthesis of new strong structures, with possible applications in nanotechnology.