Curved nanostructured materials

Graphite is a layered material that is very flexible, in which each layer is able to curve in order to form cages, nanotubes, nanocoils, nanocones, etc. In this paper, we demonstrate that various synthetic routes are capable of producing graphite-like nanomaterials with fascinating electronic and mechanical properties. There are other layered systems, which could curl and bend, thus generating novel nanostructures with positive and negative Gaussian curvature. In this context, we will also demonstrate that hexagonal boron nitride, tungsten disulfide (WS2), molybdenum disulfide (MoS2) and rhenium disulfide (ReS2) are also able to create nanocages, nanotubes and nano-arrangements exhibiting novel physico-chemical properties that could revolutionize materials science in the 21st century.

lattice constants a = 2.46 Å and c = 6.71 Å, containing two atoms in the asymmetric unit with fractional coordinates (0, 0, 0 and 2/3, 1/3, 1/5; see figure 1). As mentioned earlier, a key concept in materials is the 'curvature' of the surface. If we consider that graphite and other layered materials form surfaces, then it is possible to define curvature as if the surface was a continuum. A 2D surface embedded in a 3D space exhibits two principal curvatures C 1 and C 2 . The product of the two principal curvatures is called the Gaussian 'K ' curvature (K = C 1 C 2 ) and the average curvature, known as mean curvature, can be expressed as H = (C 1 + C 2 )/2 [9,10,15]. According to the type of Gaussian curvature K , there are three types of geometries in a 3D space [16]: (a) Euclidean geometry, where K is zero (both principal curvatures are zero); (b) spherical or ellipsoidal geometry, where K > 0 (the surface bends equally at both sides), (c) hyperbolic geometry, where K < 0 (the principal curvatures possess different signs and the surface is composed of saddle points) (figure 2).
These three geometries exhibit different properties. For example: in Euclidean geometry the sum of the interior angles of a triangle is exactly 180 • , in spherical geometries this sum is 126 greater than 180 • and in hyperbolic geometries the sum of the interior angles of a triangle is less than 180 • . Therefore, the plane is a Euclidean surface, the sphere belongs to spherical geometry and a surface composed of saddle points corresponds to hyperbolic geometry. In nature, there are structures with different levels of complexity and various types of curvature, which are closely related to mathematical complexity (see below). It is therefore clear that graphite belongs to the Euclidean geometry and C 60 to the spherical geometry, but these are not all the possibilities. Applying the Gauss-Bonnet theorem to a closed orientable surface, and Euler's law to a graphitic sheet, in which every atom is linked to three others (sp 2 bonding), it is possible to obtain the following relationship: where N 4 , N 5 , N 7 and N 8 are the number of squares, pentagonal, heptagonal and octagonal rings of carbon, respectively, and 'g' is the genus of the structure which refers to the complexity of the arrangement. As 'g' increases the complexity also increases. In other words, 'g' is the number of handles or holes in the structure; 'g = 0' for a sphere, 'g = 1' for a torus. Note that in the above equation the hexagonal rings are not present: this is because hexagons do not contribute to the topology (curvature) of the arrangement. Since C 60 has the same topology as the sphere, g = 0, then N 5 = 12 and N 4 = N 7 = N 8 = 0. Therefore, there are just 12 pentagons in C 60 (the same number of pentagons in a soccer ball).

C 60 and other fullerenes
Kroto, together with his colleagues Curl, Smalley, Heath and O'Brian, carried out experiments consisting of firing a high power laser against a rotating graphite disc under a He atmosphere in order to produce cold carbon clusters. Kroto wanted to simulate the conditions of red giant star formation; these explosions usually produce long carbon chains [17]. Using a time-of-flight mass spectrometer, Kroto and his colleagues identified a large peak commensurate with 60 carbon atoms. After intense discussions and thinking, they concluded that the most stable structure was that of a truncated icosahedron (60 vertices) and decided to call it Buckminsterfullerene or C 60 .
In these experiments, it was also found that there was a smaller peak related to a mass of 70 atoms, which later was identified as another carbon cage molecule consisting of 12 pentagons and 10 additional carbon atoms around the waist of C 60 . If C 60 (I h symmetry) is similar to a soccer ball, C 70 (D5 h symmetry) is similar to a rugby ball (figure 3). In 1990 Krätschmer, Huffman and colleagues [2] found that crystals of C 60 could be produced in macroscopic quantities using an electric arc discharge apparatus. Taylor, Harold Kroto and colleagues [18] also obtained the first NMR spectra of C 60 and C 70 and confirmed that the structure of C 60 was indeed a truncated icosahedron with I h symmetry (nano-soccer ball). The crystalline structure of solid C 60 consists of a face centred cubic (FCC) arrangement of C 60 molecules with a lattice parameter of 14.2 Å (a C 60 molecule is 7 Å in diameter; figure 4). When these fullerene crystals are doped with alkali metals, such as potassium, caesium or rubidium, it is possible to observe superconductivity as high as 33 K [19,20]. The insertion of 13 C isotopes in C 60 might shed some light on the vibrational properties of fullerenes [21], and therefore on the superconductivity mechanism.
Kroto generalized the idea of carbon cages and proposed that graphite could produce a whole family of nested cages, now known as giant fullerenes (figure 5). In this context, it is possible to distinguish between classical and non-classical fullerenes: a classical fullerene is a closed carbon cage containing 12 pentagonal rings and any number of hexagons (except one). A non-classical fullerene can admit rings with more that six carbon atoms (heptagons and octagons) and an additional number of pentagons or squares (see the section on graphitic onions).
According to equation (1), it is possible to preserve the genus zero of a fullerene (sphere) by adding rings containing more than six atoms. However additional smaller carbon rings (<6) are necessary to balance this equation and the curvature of the structures. There is experimental evidence that C 70 , C 76 [22] and C 84 [23] also form crystals. In fact, even larger fullerenes exist, but in very small quantities. The most common method used for producing C 60 is the electric arc discharge of two graphite electrodes [2] (figure 6(a)). In this experiment, the anode is consumed during the reaction, thus forming soot around the chamber; the arc temperatures are around 4200 • C. Alternatively, C 60 can also be produced via pyrolysis of hydrocarbons in inert atmospheres (naphthalene, coranulene, acetylene, etc) at lower temperatures (e.g. 1000 • C; figure 6(b)). There are 1812 mathematical ways of forming a closed cage with 60 carbon atoms (isomers), but C 60 is the most special and stable because all its pentagons are isolated by hexagons. This condition is called the isolated pentagon rule (IPR), which tends to make fullerenes more stable [24]. The next fullerene, which follows the IPR, is C 70 (it has 8149 possible isomers). Fullerenes with less than 60 atoms cannot have isolated pentagons and therefore they should be highly unstable. Nevertheless, Piskoti et al [25] claimed to have synthesized crystals of C 36 . Unfortunately, the existence of C 36 has not been fully confirmed to date.
The geometrical construction of a family of chiral fullerenes with icosahedral 'I' symmetry has been proposed by Terrones et al [26]. This new type of fullerene is energetically stable and exhibits energies similar to those presented by 'I h ' fullerenes (figure 7). In addition, these structures tend to be metallic.
Other applications of large fullerenes and graphitic polyhedral particles have been proposed for encapsulating different types of elements and compounds in their hollow cavities. In particular, radioactive material can be introduced inside, thus avoiding dangerous leaks [27,28]. In addition, the role of fullerenes in order to inhibit enzymes related to different viruses has been demonstrated [29,30].
Regarding the formation of C 60 two main mechanisms have been proposed: (i) the pentagon road [31] and (ii) the ring road [32,33]   fullerenes, with one heptagonal ring and 13 pentagons, might play an important role in the growing mechanism of C 60 . The authors found that it is possible to establish routes to grow fullerenes from C 40 to C 50 to C 60 and C 70 by the addition of carbon dimers to non-classical carbon cages (figure 8) [34]. Unfortunately, no clear experimental evidence for fullerene formation has been reported.

Onion-like structures
Iijima [35] was the first in recognizing the concentric nature of graphitic polyhedral particles in 1980 using high-resolution transmission electron microscopy (HRTEM). In 1992, Ugarte [11,12] observed that intense electron irradiation is capable of transforming amorphous carbon and graphitic polyhedral particles into quasi-spherical graphitic nanostructures, known as onion-like particles. These carbon onions can be considered as concentric (nested) arrangements of giant fullerenes, separated by 3.4 Å (the separation between layers in turbostratic graphite). Both Iijima and Ugarte suggested that fullerenes such as C 60 should be created in the centre of the onions.
It is well known that, as classical fullerenes (12 pentagons only) become larger, the structure becomes more faceted (figure 9) and not spherical like Ugarte's onions. Several groups started to address this problem suggesting the introduction of large carbon rings (heptagons and additional pentagons), so that more spherical structures are created [36]- [38]. In 1997 a mechanism explaining the transformation of polyhedral particles into quasi-spherical graphitic onions was proposed [39]. This mechanism states that fast electrons (from the electron beam generated in the microscope) displace some carbon atoms in the polyhedral particle, thus creating vacancies  in the strained regions (such as the pentagonal rings). Under these circumstances the structure loses rigidity and starts to reconstruct in order to become stable and spherical. In order to keep the sphericity, the active sites must be satisfied with heptagons and additional pentagons (figure 10). These graphitic onions represent metastable states with fascinating properties. Ab initio calculations demonstrated that quasi-spherical onions behave differently electronically when compared to standard icosahedral fullerene cages [40].
It has also been possible to introduce magnetic materials inside carbon onions [41,42], via thermolytic processes and high electron irradiation [43]. These materials may have important applications in the fabrication of magnetic inks and toners for xerography. More recently, Ajayan and Banhart [43] demonstrated that the internal shells of these onions could be transformed into diamond when high temperatures in conjunction with high electron irradiation are used.

Carbon nanotubes
Besides C 60 , the most studied carbon nanostructure is the nanotube. According to the number of layers, carbon nanotubes can be single-walled carbon nanotubes (SWCNTs) or multi-walled carbon nanotubes (MWCNTs; figure 11). The structure of CNTs were first reported in 1991 by Iijima [3]. Iijima realized that graphite could be bent to form MWCNTs with different helicities or chiralities, which refer to the way hexagonal carbon rings are arranged with respect to the tubule axis ( figure 12). The structure of CNTs can be characterized using two indices 'm' and 'n': in armchair CNTs 'm = n' (m, m), and in zigzag CNT 'n = 0' (m, 0). In chiral CNTs 'm' and 'n' are different (m, n). Hamada et al [44] demonstrated theoretically that the electronic properties of SWCNTs depended on the diameter and chirality: in particular, all of the so-called armchair type nanotubes are electronic conductors (figure 12), whereas zig-zag nanotubes are semiconductors except for the cases in which 'm − n' is a multiple of 3 ( figure 12). Indirect measurements revealed that MWCNTs possess a Young's modulus around 1.8 TPa, which is 100 times larger than that of steel [45]. For this reason, it has been regarded as a structure 100 times stronger than steel and six times lighter (weight-wise). Although composite materials containing CNTs will not be as robust as individual nanotubes, the mechanical properties could be greatly enhanced. CNTs are produced using four different methods: arc discharge of graphite electrodes in inert atmospheres [3,46], pyrolysis of hydrocarbons over catalysts [47], laser vaporization of graphite targets [48] and electrolysis of graphite electrodes in molten salts [49,50]. The arc discharge method (figure 6(a)) is very similar to that used for generating fullerenes [2] with two main differences: first, the inert gas pressures are higher (about 500 Torr of He; for producing fullerenes the pressure is about 100 Torr), and second, the nanotubes grow inside a deposit formed on the cathode, and not in the soot on the walls of the chamber. This method produces highly crystalline MWCNTs with diameters ranging from 2-30 nm. The separation between the cylinders is around 3.4 Å, which is very close to that of graphite (3.35 Å) and corresponds to turbostratic graphite. The lengths of these nanotubes can be up to 30 µm. The optimum conditions for generating nanotubes with this method consist of using currents of 150 A, voltages of 25 V and graphite electrodes between 6 and 8 mm in diameter with a separation between them of 1 mm, always under a helium atmosphere. Unfortunately, the electric arc reaction is too violent, so it is very difficult to control the homogeneity of the tubes (dimensions and shapes).
Two main formation mechanisms have been proposed for arc-grown CNTs: the first establishes that carbon units (C, making the tube grow until certain instabilities in the arc produce defects, such as pentagonal rings, causing its closure [51]. The second mechanism proposes that CNTs are elongated fullerenes that grow by the addition of carbon atoms to their surface [52]. It is noteworthy that, besides CNTs, other structures are produced, such as polyhedral particles and amorphous carbon. Therefore, it is important to purify the material obtained by oxidation at high temperatures [46] or by dispersion of colloidal suspensions with surfactants [53,54]. An analysis of the competition of single-and multi-walled tube formation against fullerenes and onion structures, based on the difference of surface and bond bending energies, has also been reported [55]. With the pyrolysis method it is possible to acquire more control in the CNT formation. This method consists of heating a hydrocarbon or an organic precursor containing carbon in the presence of a metal catalyst (e.g. nickel, cobalt or iron; figure 6(b)). Before the discovery of C 60 , it was well known that carbon fibres could be obtained by pyrolytic methods using hydrocarbons [56,57]. In fact, Endo reported the first image of a 4 nm SWNT in the mid-1970s [55]. However, in general, carbon fibres are not nanotubes since the fibres, which usually do not consist of concentric graphene tubes, are larger in diameter and exhibit a great amount of graphitic defects and impurities. Nevertheless, the application of carbon fibres is widespread, showing a mechanical strength comparable to that of steel. According to Dresselhauss et al [56] the first record of carbon fibres was realized by Schützenberger and Schützenberger in 1890, more than 100 years ago. Two mechanisms have been proposed for the formation of carbon fibres which can be extended to nanotubes: the first, proposed by Baker et al, consists of the diffusion of carbon through the catalytic particle depositing the carbon material on the other side, forming the fibre [56]. in which the filaments are formed by diffusion of carbon around the catalytic particle's surface, generating the fibre [56] (chapter 2 of [55] gives a complete account of the synthesis of graphite fibres and filaments). In either of these mechanisms the size of the particle, the nature of the hydrocarbon and the temperature play a crucial role [56,57].
In 1994, Ajayan and colleagues [58] found that CNTs could be aligned by embedding them in a polymer, thus forming a composite material. When the polymer was cut, the authors observed an aligned arrangement of CNTs. In this context, a better alternative is to use laser ablation of cobalt thin films and then to pyrolyse them with an organic precursor such as 2-amino 4,6-dichloro-S-tryazine [47]. The nanotubes obtained in this way are aligned in bundles which can grow up to 60-100 µm, with diameters between 30 and 50 nm (see figure 13).
The laser vaporization method involves firing a high power laser towards a graphite target inside a furnace at 1200 • C [59,60]. The condensation of the material generated by the laser is responsible for the nanotube formation. If we add nickel or cobalt to the graphite target, we obtain SWCNTs [48]. They exhibit diameters around 14 Å and form a two-dimensional crystal with a lattice constant of 17 Å. Arrangements of SWCNTs can also be obtained by electric arc discharge when arcing mixtures of carbon, nickel and yttrium [61]- [65].
The electrolysis method is based on using graphite electrodes inside a molten salt such as lithium chloride (700 • C) under an argon atmosphere [49,50]. Depending on the conditions, up to 20%-30% of the material could consist of MWCNTs. The depth of the cathode and the current (3-10 A) also play an important role in the formation of nanotubes using this technique.

Mechanical and electronic properties of carbon nanotubes
HRTEM studies reveal that CNTs are extraordinarily flexible and do not break upon mechanical deformation. For example, they can be bent mechanically by mechanical milling, or by embedding the tubes in a polymeric resin [58], [66]- [68]. Theoretical calculations predict this flexibility [69]- [71]. The first attempt to determine the stiffness of CNTs was carried out by Treacy and colleagues. Using a HRTEM to measure the amplitudes of vibrating tubes at different temperatures, the authors calculated indirectly Young's modulus [45,72]. They found that MWCNTs exhibit Young's moduli of the order of 1.8 TPa, which is higher than conventional carbon fibres (about 800 GPa). Direct measurements using atomic force microscopy (AFM) revealed that the Young's modulus of MWCNTs is around 1.28 TPa [72,73]. This high Young's modulus makes CNTs a very strong material. However, the challenge of building a super-strong composite material out of nanotubes is still underway.
The first measurements of the electrical conductivity of MWCNTs were recorded using gold micro-contacts attached to the tubes by lithographic techniques. These showed that the resistance depends on the temperature and that in the range between 2 and 300 K the tubes were semimetallic [73]. Conductivity measurements on aligned MWCNTs show that the material behaves as a nanoconductor [74,75]. For SWCNTs, it was confirmed that, depending on the diameter and chirality, CNTs could be metallic conductors or semiconductors [76,77]. SWCNTs around 14 Å in diameter also exhibit quantum (discrete) electronic conductivity [76]. It is noteworthy that the atomic arrangement (the geometry of the structure) determines the mechanical and electronic properties of CNTs. Nowadays, the production of CNT crystals has been reported [78]. Unfortunately, these experiments have not been reproduced by other groups.
It has been observed (in situ) that SWCNTs can coalesce under high electron irradiation and high temperature conditions (around 800 • C, figure 14) [79]. Tight binding molecular dynamic (TBMD) calculations suggest that vacancies, generated by knock-on displacements of carbon atoms by fast electrons inside the electron microscope, trigger this coalescence [79]. Moreover, in order to achieve coalescence, the tubes should possess the same chirality, otherwise tube polymerization might take place. Recent experiments by Terrones and colleagues [80] have shown that CNT junctions can be produced under similar conditions (see figure 15); this opens up the possibility of synthesizing CNT networks and novel resistant and lightweight carbon fabrics.

Boron nitride nanotubes
hBN is another layered material with a structure very similar to graphite, having hBN ring layers separated by 3.33 Å, in which every boron is connected to three nitrogen atoms and vice versa; the B-N distance is 1.44 Å. Between the layers, every boron interacts with a nitrogen atom through a van der Waals force (figure 16). Bulk hBN is an insulator with a bandgap of 5.8 eV [81], whereas graphite is a semimetal (band overlap = 0.04 eV) [57]. The first groups who reported the existence of BN nanotubes used the electric arc discharge method with high melting point metal (tantalum and tungsten, for example) electrodes and BN powder inside the anode [82]- [84]. Loiseau et al [85] reported the production of BN nanotubes by arcing HfB 2 electrodes in an inert atmosphere. This technique led to the formation of mainly single-and double-walled BN tubes.
It was noticed that, in BN nanotubes, the tips were squared, meaning that instead of having pentagonal rings to close the structure (as in CNTs), there are BN squares. This is energetically more favorable because a pentagon will exhibit B-B or N-N bonds, which are more unstable. In terms of curvature, BN admits, with high stability, squares for the positive curvature and octagons for the negative curvature. Nevertheless, Fowler and colleagues have found using density functional tight binding (DFTB) calculations that nitrogen excess in BN cages is stable [86]. Theoretical studies revealed that BN nanotubes are insulators with a bandgap of 5.5 eV [87], a bit smaller than in bulk BN. This property is independent of tube diameter, chirality and number of walls. BN nanotubes may also be extremely resistant to oxidation and have exhibited high Young's moduli (see below). Therefore, the tubes may find important uses in ceramics and composites.
Han et al [88] reported the efficient generation of BN nanotubes by reacting pyrolytically grown CNTs with B 2 O 3 in the presence of N 2 at high temperatures (1300-1500 • C). Subsequent studies by Golberg et al [89] have shown that BN nanotube bundles can also be produced in bundles when CNTs are exposed to mixtures of B 2 O 3 , V 2 O 5 and MoO 3 in a N 2 atmosphere at elevated temperatures. Very recently, alternative experimental approaches have been developed in order to generate BN nanotubes and other BN nanostructures: (a) thermolysis of borazine (B 3 N 3 H 6 ) over nickel boride catalyst particles at 1000-1100 • C [90]; (b) thermal annealing of powdered rhombohedral boron and hBN mixtures at 1200 • C in lithium vapour [91]; (c) plasma jets [92,93], and (d) ball milling of hBN, followed by thermal annealing at 1300 • C [94,95].
At this stage, novel composites and nanoscale components need to be fabricated employing BN nanotubes, which appear to be extremely resistant to oxidation. Experimental determination of the Young's modulus of BN tubes (Y = 1.22 ± 0.24 TPa) [96] confirms the values obtained using theoretical approaches for BN nanotubes [97,98] and reveals that BN tubes are highly crystalline and may be the strongest insulating nanofibres. Graphite and hBN have opened new possibilities for producing novel nanomaterials with different types of B x C y N z systems [99].
Pyrolysis experiments with melamine (trisaminotriazine) have been helpful in synthesizing C x N y nanofibres: Terrones et al [100,101] described nanotubes with stoichiometries close to C 12 N and confirmed that nitrogen is bonded in a sp 2 fashion within the carbon framework.
Boron-doped CNTs have also been synthesized by electric arc discharge with hBN inside the graphite anode [102]. In this experiment it is noticed that CNTs are highly graphitic and also very long, around 20-100 µm. It was found that B was usually located at the tips of the tubes and that the zig-zag chirality was preferred among tubes [102]. Blase and colleagues found using molecular dynamics ab initio calculations that boron was acting as a surfactant, avoiding the zig-zag tubes closure [102]. In addition, boron-doped CNTs exhibit conducting properties.
On the other hand, nitrogen-doped CNTs can also exhibit fascinating features such as pyridine-like species within the graphitic mesh (figure 17) [103], which causes a peak close to the Fermi level in the local density of states (LDOS). All these nanotubes, armchair and zig-zag are found to be metallic. Therefore, the nitrogen is donating electrons to the nanotube system in order to enhance electronic transport.

Metal dichalcogenide curved nanostructures
Another type of layered material that can be bent are molybdenum and tungsten disulfides (MoS 2 and WS 2 ). The most stable forms of MoS 2 and WS 2 consist of a metal layer sandwiched between two sulfur layers (space group P63/mmc). These triple layers are stacked together, similar to graphite, by van der Waals interactions separated by 6.15 Å for MoS 2 and 6.18 Å for WS 2  Reshef Tenne's group found that MoS 2 and WS 2 could form closed cages (inorganic analogues of fullerenes) and nanotubes by substitution reactions under H 2 S atmospheres [104]- [106]. These nanostructures have proved to be exceptional solid lubricants [107]. DFTB calculations demonstrated that both armchair and zig-zag MoS 2 and WS 2 are semiconductors, with a gap decreasing as the diameter decreases. In bulk, flat MoS 2 and WS 2 layers exhibit a semiconducting behaviour with a larger gap than that observed for the respective nanotubes [108,109]. Interestingly, armchair nanotubes exhibit indirect and direct gaps, similar to the bulk material, whereas zig-zag tubes only display direct gaps. Nevertheless, all MoS 2 and WS 2 nanotubes will be semiconducting ( figure 18).
In order to close MoS 2 and WS 2 layers, preserving the stoichiometry, square-like defects (SLD) and octagonal-like defects (OLD) are needed. In figure 19 we show how to achieve closure in zig-zag and armchair disulfide nanotubes. Note that the positive curvature is given by the SLD and the negative curvature by the OLD. Inorganic fullerenes will need 6 SLD to close the cage [109] ( figure 19). Rao's group [110] reported the existence of NbS 2 and TaS 2 nanotubes by hydrogen reduction of trisulfides NbS 3 and TaS 3 at 700 • C; in this work, there are also nanoribbons. DFTB calculations on NbS 2 nanotubes reveal that all of them are metallic [111]. The metallic character of these tubes arises from the fact that Nb has one d electron less than Mo or W in the valence shell. In fact, the flat layer of NbS 2 is metallic and becomes a superconductor around 6-7 K.    arc discharge with a metal powder inside the anode, one could produce CNTs filled with metal carbides [68], [114]- [116]. A difficulty with the method mentioned above is that the tubes are only partly filled; this might affect the conductivity properties. Loisseau and Pascard [117] found that nanotubes could be filled significantly when small amounts of sulfur are involved in the electric arc experiments. Moreover, there are chemical techniques (using nitric acid treatments), in which the tubes can be opened and nanomaterials, such as oxides of Pd, Ag, Au, Co, Fe, Ur, Ni, Mo, Sn, Ny, Eu, La, Ce, Y, Zr and Cd, could be inserted into their interior [118]- [122]. It is also important to mention that enzymes and proteins have been introduced inside nanotubes [123,124]. On the other hand, capillarity effects have been used to introduce elements inside CNTs such as Sn, Pb, Bi, Cs, S and Se [125]- [127]. Pyrolysis experiments with ferrocene (FeCp 2 ) and C 60 have proved to be very useful to produce aligned Fe-filled carbon MWCNTs [128]. Magnetic measurements on these tubes indicate that they possess higher magnetic coercivities than those reported for nickel and cobalt nanowires [128]. It is noteworthy that the graphene layers avoid oxidation, protecting very well these ferromagnetic nanowires. It is believed that Fe nanowires might find applications as magnetic storage devices. Recently, it has been found that spray pyrolysis of mixtures of Figure 25. Quasi-periodic graphitic structure with I h symmetry oriented along the five-fold axis. Table 3. Schwarzite G8bal name, space group, lattice parameter, number of atoms in the cubic cell and fractional coordinates are given. The genus per primitive cell is 3. This structure divides space into two equal regions. nickelocene (NiCp 2 ) and ferrocene in benzene produce nanowires with a Ni-Fe alloy called invar (Fe 65 Ni 35 ) [129]. In the future, we expect that pyrolysis techniques will enable more control of the filling of the nanotube in order to produce nanowires with exact alloy compositions. An alternative to filling MWCNTs is by electrolysis of graphite electrodes in molten salts containing small amounts of tin, bismuth or lead chloride [130].    It is important to note that other types of inorganic materials, fullerenes and gases have also been incorporated inside the cylindrical cores of SWCNTs and MWCNTs. However, this topic is beyond the scope of this review.

Carbon nanocones
Theoretical studies predicted the formation of graphitic cones [131,132]. Subsequently, isolated graphitic cones were produced by carbon condensation on a graphite substrate [133] and by pyrolysis of heavy oils [134]. More recently, single-walled aggregates of conical graphitic structures have been prepared by laser ablation of graphite targets [135]. Conical structures consisting of other layered materials such as BN have also been prepared by reacting boron oxide vapours with MWCNTs [136].
Very recently, it was found that pyrolysis of palladium precursors always produces conical nanofibres (figure 21) [137]. An important feature of these new nanostructures is that they are held together by van der Waals forces since the fibre is composed of an arrangement of stacked cones, which could be opened or closed ( figure 22). It is believed that nanocones may be good electron field emitters. Charlier's group has calculated the electronic properties of nanocones, finding that there is a charge accumulation towards the tip and that these are localized states near the Fermi level (E F ). These features make them suitable as field emitters [138].  In 1991, Mackay and Terrones [13] proposed that the inclusion of carbon rings with more than six atoms in a graphite hexagonal mesh could produce stable periodic graphitic arrangements with negative curvature analogous to zeolites (figure 23). These new hypothetical arrangements, called 'Schwarzites', exhibit topologies similar to the TPMS [9,10], [13]- [15], which have attracted mathematicians since the 1890s (see figure 24). TPMS possess, at all their points, zero mean curvature, so H = 0 everywhere. Schwarzites and TPMS show labyrinths that can divide space into two congruent regions. Energetic calculations have been performed in Schwarzites; they are more stable than C 60 [139]- [142]. This stability comes from the fact that in the heptagonal or octagonal rings there is very little mechanical strain, thus preserving the sp 2 -like nature of graphite (120 • angles between nearest neighbours). As we have observed earlier, the heptagons or octagons generate negative curvature or saddle points necessary for periodicity and complexity. The genus per primitive cell of the most common Schwarzites is 3 and 4 [143], so they are more complex than fullerenes, which possess genus 0 (see equation (1)).

126.33
Among the possible applications of Schwarzites we can suggest: semiconducting nanodevices, new catalysts and molecular sieves. Heptagonal rings of carbon not only produce periodicity, but also can generate quasi-periodicity or disorder in a graphitic architecture [144] (see figure 25). Fractional coordinates and characteristic features of Schwarzites are given in tables 1-7.
We have seen that complexity and periodicity are linked with the number of rings (higher than six carbon atoms; see equation (1)). One question emerges here: can we propose a closed complex structure (genus greater than zero) without periodicity? The answer is yes. Itoh, Ihara and Kitakami have shown that toroidal graphite is stable. These tori exhibit genus 1 (g = 1) possessing pentagons, hexagons and heptagons [145,146] (see figure 26). Also, helicoidal graphitic tubes have been found experimentally [147]. However, the most complex hypothetical arrangements have been proposed by Terrones and Terrones [144]. These structures lack pentagonal rings and contain only heptagons and hexagons. The genus of these structures goes up to g = 21, having the same topology as a sphere with 21 handles (figures 27 and 28). An important feature of the complex graphitic structures is that they exhibit holes or labyrinths, in which molecules can be inserted. TB calculations showed that around the holes (necks), the electronic behaviour is metallic [148].

Haeckelites
The presence of defects such as pentagons and heptagons in fullerenes modifies the electronic properties. A new hypothetical type of graphene sheet, which admits pentagons, heptagons and hexagons, has been proposed, noting that the number of heptagons and pentagons should be the same in order to compensate for the negative curvature of the heptagons and the positive curvature of the pentagons [149] (figure 29). These arrangements are now called 'Haeckelites' in honour of Ernst Haeckel, a German zoologist who produced a beautiful drawing of radiolaria (micro-skeleta of zoo-plankton), in which heptagonal, hexagonal and pentagonal rings were observed [150]. An important property of Haeckelites is that they all exhibit a significant electronic signal at E F in the LDOS. This means that all these systems should possess metallic behaviour. Thus, it is possible to roll up Haeckelite sheets to form nanotubes, which will be conductors, independent of the diameter and chirality.
Another property of Haeckelite tubes is that they keep the stiffness of classical CNTs, composed of only hexagons; the Young's modulus of Haeckelite tubes is around 1.0 TPa. In addition, Haeckelites also exhibit local rugosity due to the local curvature introduced by the presence of heptagons and pentagons (figure 30). It might be possible that electron irradiation experiments generate heptagon-pentagon pairs (rugosity) in SWCNTs. For fractional coordinates of Haeckelites see tables 8-10.

Conclusions
Different layered materials could curve and bend in order to form various nanostructures with novel properties. Graphite is just one example of a planar structure that can be transformed into fullerenes, nanotubes (multi-walled or single-walled), nanocones, etc. However, hBN, WS 2 , MoS 2 , NbS 2 and ReS 2 could also be bent in order to form new nanostructures. Computer simulations and experimental work indicate that curvature, introduced via different types of 'defects', produces stable atomic arrangements. These 'defects', which can be pentagons, heptagons, rings with more atoms, or even more complex architectures, modify the electronic, magnetic and mechanical properties of the systems. Along this line, doping of curved nanostructures is a promising area of research. We are at the beginning of a new field in which new curved layered nanomaterials are waiting to be discovered. At present the main challenge is to control their production and to generate larger amounts with specific characteristics so that the fabrication of novel nanotechnological devices becomes a reality in the twenty-first century.