Diamond D₅, a Novel Class of Carbon Allotropes

Abstract

Design of hypothetical crystal networks, consisting of most pentagon rings and generically called diamond D₅, is presented. It is shown that the seed and repeat-units, as hydrogenated species, show good stability, compared with that of C₆₀ fullerene, as calculated at DFT levels of theory. The topology of the network is described in terms of the net parameters and Omega polynomial.

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11.1 Introduction

The nano-era, a period starting since 1985 with the discovery of C₆₀, is dominated by the carbon allotropes, studied for applications in nano-technology. Among the carbon structures, fullerenes (zero-dimensional), nanotubes (one dimensional), graphene (two dimensional), diamond and spongy nanostructures (three dimensional) were the most studied (Diudea 2005; Diudea and Nagy 2007). Inorganic compounds also attracted the attention of scientists. Recent articles in crystallography promoted the idea of topological description and classification of crystal structures (Blatov et al. 2004, 2009; Delgado-Friedrichs and O’Keeffe 2005).

Diamond D₆ (Fig. 11.1), the classical, beautiful and useful diamond has kept its leading interest among the carbon allotropes, even as the newer “nano” varieties. Along with electronic properties, the mechanical characteristics appear of great importance, as the composites can overpass the resistance of steel or other metal alloys. A lot of efforts were done in the production and purification of “synthetic” diamonds, from detonation products (Decarli and Jamieson 1961; Aleksenskiǐ et al. 1997; Osawa 2007, 2008) and other synthetic ways (Khachatryan et al. 2008; Mochalin and Gogotsi 2009).

Fig. 11.1

Diamond, a triple periodic network: Ada(mantane) D₆_10_111 (left), Dia(mantane) D₆_14_211 (central) and Diamond D₆_52_222 net (right)

However, the diamond D₆ is not unique: out of the classical structure, showing all-hexagonal rings of sp³ carbon atoms in a cubic network (space group Fd3m), there is Lonsdaleite (Frondel and Marvin 1967; He et al. 2002) a rare stone of pure carbon discovered at Meteor Crater, Arizona, in 1967 and also several hypothetical diamond-like networks (Sunada 2008; Diudea et al. 2010). The Lonsdaleite hexagonal network (space group P6₃/mmc) is illustrated in Fig. 11.2.

Fig. 11.2

Losdaleite, a double periodic network: L₆_12_111 (left), L₆_18_211 (central) and L₆_48_222 net (right)

Dendrimers are hyper-branched nano-structures, made by a large number of (one or more types) substructures called monomers, synthetically joined within a rigorously tailored architecture (Tomalia et al. 1990; Newkome et al. 2001; Diudea and Katona 1999). They can be functionalized at terminal branches, thus finding a broad pallet of applications in chemistry, medicine, etc.

Multi-tori MT are structures of high genera (Diudea 2010a), consisting of more than one tubular ring. They are supposed to result by self-assembly of some repeat units (i.e., monomers) which can be designed by opening of cages/fullerenes or by appropriate map/net operations. Multi-tori, rather than dendrimers, appear in processes of self-assembling of some rigid monomers. Zeloites and spongy carbon, recently synthesized (Barborini et al. 2002; Benedek et al. 2003) also contain multi-tori.

Structures of high genera, like multi-tori, can be designed starting from the Platonic solids, by using appropriate map operations (Diudea 2010a). Such structures have before been modeled by Lenosky et al. (1992) and Terrones and Mackay (1997) etc.

11.2 Dendrimer Design and Stability

A tetrapodal monomer M₁(Fig. 11.3, left), designed by Trs(P ₄(T)) sequence of map operations (Diudea and Ilić 2011) and consisting of all pentagonal faces, can self-arrange to a dendrimer M₅, at the first generation stage (Fig. 11.3, right).

Fig. 11.3

Tetrapodal unit designed by Trs(P ₄(T)) and the corresponding dendrimer, at first generation stage

The “growing process” is imagined occurring by identification of the trigonal faces of two opposite M₁ units; at the second generation, six pentagonal hyper-cycles are closed, as in molecule M₁₇, Fig. 11.4 (Diudea and Ilić 2011).

Fig. 11.4

Dendrimer at second (left) and fifth (right) generation stage; M₅₇ = 4S_MT; v = 972; e = 1770; f ₅ = 684; g = 58 (infinite structure); adding f ₃ = 40, then g = 38 (finite structure)

The process is drown as a “dendrimer growth”, and is limited here at the fifth generation (Fig. 11.4), when a tetrahedral array results: 4S_MT = M₅₇.

Multi-tori herein considered can be viewed either as infinite (i.e., open) structures or as closed cages; then, it is not trivial to count the number of simple tori (i.e., the genus g) in such complex structures.

The Euler’s formula [Euler (1758)]: $v - e + f = 2(1 - g)$, where v, e and f are the number of vertices/atoms, edges/bonds, and faces, respectively, is applicable only in case of single shell structures. In multi shell structures (Diudea and Nagy 2008), have modified the Euler formula as: $v - e + r - p(s - 1) = 2(1 - g)$, where r stands for the number of hard rings (i.e., those rings which are nor the sum of some smaller rings), p is the number of smallest polyhedra filling the space of the considered structure while s is the number of shells. In case of an infinite structure, the external trigonal faces are not added to the total count of faces/rings. The calculated g-values are given in Fig. 11.4.

The number of tetrapodal monomers, added at each generation, up to the fifth one, realized as M₅₇, is: 1; 4; 12, 24, 12, 4. The connections in M₅₇ are complex and to elucidate the large structures up to the fifth generation, the design of the corresponding reduced graphs (Fig. 11.5) was needed (Diudea 2010b, 2011).

Fig. 11.5

Reduced graphs at 2nd (left) and 5th (right) generation stage; C₅₇: v = 57; e = 94; r ₅ = 42; g = 0.5; R(C₅₇,x) = 42x⁵ + 82x⁹ + 144x¹⁰

The structure C₁₇ (Fig. 11.5, left) we call the “seed” of all the hereafter structures. The structure C₅₇ (Fig. 11.5, right) corresponds to the above M₅₇ and is equivalent to 4 “condensed” dodecahedra, sharing a common point. By considering this common point as an internal shell s, the modified (Diudea and Nagy 2008) Euler formula will give (for v = 57; e = 94; r = 42; p = 4 and s = 2) a (non-integer) genus g =0.5. The ring polynomial R(x) is also given, at the bottom of Fig. 11.5.

11.3 Diamond D₅ Networks

11.3.1 Spongy D₅

A monomer C₈₁, derived from C₅₇ and consisting of four closed C₂₀ units and four open units, and its mirror image (Fig. 11.6) was used by (Diudea and Nagy 2011a) to build the alternant network of spongy diamond SD₅ (Fig. 11.7). The nodes of diamond SD₅ network consist of C₅₇ units and the network is triple periodic.

Fig. 11.6

Monomer C₈₁ unit (left-up), and its mirror image-pair (right-up); the monomers as in the triple periodic network of diamond SD₅

Fig. 11.7

SD₅ (C₅₇) triple periodic network: top view (left) and corner view (right)

The number of atoms v, bonds e, and C₅₇ monomers m, and the content in sp ³ carbon, given as a function of k – the number of monomers along the edge of a cubic (k,k,k) domain. At limit, in an infinitely large net, the content of sp ³ carbon approaches 77% (see Appendix 2, Table 11. 4). The density of the net varies around an average of d = 1.6 g/cm³, in agreement with the “spongy” structure of D₅ net (Fig. 11.7).

A possible pathway to D₅ was proposed by Diudea and Nagy (2011a) (Fig. 11.8):

Fig. 11.8

Pathway to D₅

The main intermediate structures in this scheme are: C₁₇ (the “seed” of D₅), C₄₁, C₅₃ and C₈₁(5) (see Fig. 11.8). The stability of these structures was evaluated as hydrogenated species. The C₈₁ monomer has a C₅₇ core and contains 12 flaps which represent half of the junctions between the SD₅ nodes.

To avoid the non-wished side products C₆₅(6) to C₈₁(6) (containing six-membered rings), the suggested way is through C₅₃. In a next step, one can reach either C₅₇ as a final structure (which can, however, lead to some dense species of D₅) or go to C₈₁(5), the monomer of SD₅ network.

Of course, the scientist will choose the most convenient route in an attempt to synthesize these structures.

11.3.2 Dense D₅

There is a chance to reach D₅ just from C₁₇, a centrohexaquinane (Paquette and Vazeux 1981; Kuck 2006) which can dimerize (Diudea and Nagy 2011b; Eaton 1979) to 2 × C₁₇ = C₃₄ and this last condensing to 4 × C₁₇ = C₅₁ (Fig. 11.9, top row).

Fig. 11.9

Way to Ada_20: 2 × C₁₇ = C₃₄ (top-left and central), 4 × C₁₇ = C₅₁ (top-right), 3(4 × C₁₇) = C₁₁₉ (bottom-left) and Ada_20_170 (bottom-right)

A linear 4 × C₁₇ = C₅₇ is also energetically possible (see Table 11.1). The angular tetramer 4 × C₁₇ = C₅₁ will compose the six edges of a tetrahedron to form the corresponding Adamantane-like Ada_20_170, bearing six pentagonal wings (in red – Fig. 11.9, right, bottom), or without wings, as in Ada_20_158 (Fig. 11.10, left). Compare this with Adamantane (Fig. 11.1, left) in the structure of classical diamond D₆. In the above symbols, “20” refers to C₂₀, which is the main unit of the dense diamond D₅ (Figs. 11.10–11.12) while the last number counts the carbon atoms in the structures.

Table 11.1 Single point calculation results (HOMO-LUMO gap in eV and total energy E_tot in a.u.) at the B3LYP/6-31 G(d,p) levels of theory; C₆₀ is taken as reference structure

Table 11.2 Omega polynomial in Diamond D₆ and Lonsdaleite L₆ nets, function of the number of repeating units along the edge of a cubic (k,k,k) domain

Table 11.3 Examples, omega polynomial in diamond D₆ and lonsdaleite L₆ nets

Table 11.4 Topology of the spongy SD₅_20

Fig. 11.10

Adamantane-like structures: Ada_20_170 (left) and Ada_28_213 (right)

A diamantane-like unit is evidenced, as in Fig. 11.11 (see for comparison the diamantane, Fig. 11.1, central). Since any net has its co-net, the diamond D₅_20 net has the co-net D₅_28 (Fig. 11.12, right), of which corresponding units are illustrated in Figs 11.10 (right)–11.11 (right), respectively. In fact is one and the same triple periodic D₅ network, built up basically from C₂₀ and having as hollows the fullerene C₂₈ (Diudea and Nagy 2011b). The co-net D₅_28 cannot be reached 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-like unit. The C₂₈-based hyperdiamond reported by Bylander and Kleinman (1993) consists of only C₂₈ fullerenes joined by the tetrahedrally disposed neightbors by only one covalent bond.

Fig. 11.11

Diamantane-like structures: Dia_20_226 net (left) and Dia_28_292 co-net (right)

Fig. 11.12

Diamond D₅_20_860_333 net (left) and D₅_28_ 1022_333 co-net (right)

Our D₅_20/28 hyperdiamond mainly consists of sp³-bonded carbon atoms building Ada-like repeating units (including C₂₈ as hollows), the cohesive bonds of these units being the same sp³ covalent (and the same distributed) bonds as within the repeating units. This is in high contrast to the (Bylander and Kleinman 1993) hyperdiamond. The ratio C-sp³/C-total trends to 1 in a large enough network. The topology of the D₅ networks is detailed in the last section.

Since the smallest C₂₀ fullerene is the highest reactive one, it is expected to spontaneously stabilize in an sp³-crystalline form, e.g., bcc-C₂₀ (Chen et al. 2004; Ivanovskii 2008) or D₅_20/28 (Diudea and Nagy 2011b). A similar behavior is expected from the C₂₀-derivative 2 × C₁₇ = C₃₄ (Fig. 11.9) to provide the Ada-like repeating units of the proposed D₅-diamond.

As the content of pentagons R[5] per total rings trend to 90% (see Table 11.5, entry 9) we called this, yet hypothetical carbon allotrope, the diamond D₅. Since the large hollows in the above spongy diamond are not counted, and the small rings are all pentagons, we also called it (S)D₅.

Table 11.5 Omega polynomial in Diamond D₅_20 net function of k = no. ada_20 units along the edge of a cubic (k,k,k) domain

Table 11.6 Examples, omega polynomial in D₅_20 net

Table 11.7 Omega polynomial in D₅_28 co-net function of k = no. ada_20 units along the edge of a cubic (k,k,k) domain

Table 11.8 Examples, omega polynomial in D₅_28 co-net

Table 11.9 Omega polynomial in Lonsdaleite-like L₅_28 and L₅_20 nets function of k = no. repeating units along the edge of a cubic (k,k,k) domain

The presence of pentagons in diamond-like fullerides and particularly the ratio R[5]/R[6] seems to be important for the superconducting properties of such solid phases (Breda et al. 2000). In D₅ this ratio trends to 9 (see Table 11.5).

Energetic data, calculated at the DFT level in Table 11.1 – (Diudea and Nagy 2011a, b) show a good stability of the start and intermediate structures. Limited cubic domains of the D₅ networks have also been evaluated for stability, data proving a pertinent stability of the (yet) hypothetical D₅ diamond.

The calculated data show these structures as energetic minima, as supported by the simulated IR vibrations. All-together, these data reveal the proposed structures as pertinent candidates to the status of real molecules.

Density of the D₅ networks was calculated with the approximate (maximal) volume of a cubic domain. The values range from 1.5 (SD₅) to 2.8 (D₅).

11.4 Lonsdaleite L₅_28 Network

By analogy to D₅_20/28, a lonsdaleite-like net was built up (Fig. 11.13).

Fig. 11.13

Losdaleite, a double periodic network: L₅_28_134 (left), L₅_28_134 (top view, central) and L₅_28_250 (side view, right)

As a monomer, the hyper-hexagons L₅_28_134 (Fig. 11.13, left and central), in the chair conformation, of which nodes represent the C₂₈ fullerene, was used. Its corresponding co-net L₅_20 was also designed. The lonsdaleite L₅_28/20 is a double periodic network, partially superimposed to the D₅_20/28 net (Diudea and Nagy 2011b).

11.5 Omega Polynomial

In a connected graph G(V,E), with the vertex set V(G) and edge set E(G), two edges e = uv and f = xy of G are called codistant e co f if they obey the relation [John et al (2007)]:

$$ d(v,x) = d(v,y) + 1 = d(u,x) + 1 = d(u,y) $$

(11.1)

which is reflexive, that is, e co e holds for any edge e of G, and symmetric, if e co f then f co e. In general, relation co is not transitive; if “co” is also transitive, thus it is an equivalence relation, then G is called a co-graph and the set of edges $C(e): = \{\, f \in E(G);\;f\;co\;e\}$ is called an orthogonal cut oc of G, E(G) being the union of disjoint orthogonal cuts: $E(G) = {C_1} \cup {C_2} \cup... \cup {C_k},\;\;{C_i} \cap {C_j} = \oslash, \;i \ne j$. Klavžar has shown (Klavžar 2008) that relation co is a theta Djoković-Winkler relation (1973–1984).

We say that edges e and f of a plane graph G are in relation opposite, e op f, if they are opposite edges of an inner face of G. Note that the relation co is defined in the whole graph while op is defined only in faces. Using the relation op we can partition the edge set of G into opposite edge strips, ops. An ops is a quasi-orthogonal cut qoc, since ops is not transitive.

Let G be a connected graph and ${S_1},{S_2},...,{S_k}$ be the ops strips of G. Then the ops strips form a partition of E(G). The length of ops is taken as maximum. It depends on the size of the maximum fold face/ring F_max/R_max considered, so that any result on Omega polynomial will have this specification.

Denote by m(G,s) the number of ops of length s and define the Omega polynomial as (Diudea 2006; Diudea et al. 2008; Diudea and Katona2009):

$$ \Omega (G,x) = \sum\nolimits_s {m(G,s) \cdot {x^s}} $$

(11.2)

Its first derivative (in x = 1) equals the number of edges in the graph:

$$ \Omega ^{\prime}\left( {G,1} \right) = \sum\nolimits_s {m\left( {G,s} \right)} \cdot s = e = \left| {E(G)} \right| $$

(11.3)

On Omega polynomial, the Cluj-Ilmenau [John et al (2007)] index, CI = CI(G), was defined:

$$ CI(G) = \{ {[\Omega ^{\prime}(G,1)]^2} - [\Omega ^{\prime}(G,1) + \Omega ^{\prime\prime}(G,1)]\} $$

(11.4)

This counting polynomial found utility in predicting stability of small fullerenes (Diudea 2010a) and in description of various polyhedral nanostructures.

11.5.1 Topology of Diamond D₆ and Lonsdaleite L₆ Nets

Topology of the classical diamond D₆ and Lonsdaleite L₆ are listed in Table 11.2. Along with Omega polynomial, formulas to calculate the number of atoms in a cuboid of dimensions (k,k,k) are given. In the above, k is the number of repeating units along the edge of such a cubic domain. One can see that the ratio C(sp³)/v(G) approaches the unity; this means that in a large enough net almost all atoms are tetra-connected, a basic condition for a structure to be diamondoid. Examples of calculus are given in Table 11.3.

11.5.2 Topology of Spongy Diamond SD₅

In describing the topology of the spongy diamond SD₅, we considered only the pentagons, the larger hollows being omitted. Thus, only the basic data of the net are presented in Table 11.4; the ratio C(sp³)/v(G) is here far from unity, because of many carbon atoms are exposed to exterior.

11.5.3 Topology of Dense Diamond D₅ and Lonsdaleite L₅

Topology of the dense diamond D₅ and lonsdaleite L₅ is presented in Tables 11.5–11.10: formulas to calculate Omega polynomial, number of atoms, number of rings and the limits (at infinity) for the ratio of sp³ C atoms over total number of atoms and also the ratio R[5] over the total number of rings (Table 11.5). Numerical examples are given.

Table 11.10 Examples, omega polynomial in L₅_28 and L₅_20 nets

11.6 Conclusions

A novel class of (hypothetical) carbon allotropes, consisting mostly of pentagon rings (going up to 90% in the total number of pentagon/hexagon rings), was here presented. The seed of these allotropes, C₁₇ and the adamantane-like repeating-units, as hydrogenated species, show a good stability, comparable with that of C₆₀ fullerene, as calculated at DFT levels of theory. The main representatives of these allotropes are the diamond D₅ and lonsdaleite L₅, in fact hyper-structures corresponding to the classical diamond D₆ and lonsdaleite L₆. The topology of the networks was described in terms of the net parameter k and Omega polynomial.