Schwarzite

Their quotient surface, under the translation group, is a surface of genus greater or equal to three. Schwarzits, i.e. ( 6,7,8, 3)-maps on surface of genus three, have been used to model 3-periodic minimal surfaces (see [KiOO]). 

Lenosky, Gonze, Teter and Elser propose the name schwarzites, in memory of H. A. Schwarz, for this category of graphites with non-positive curvatures. Since the name seems vacant as a mineral name, we commend the proposal. 

Once the average ring-six exceeds six, hyperbolic carbon structures must result, which lie on periodic h)q)erbolic surfaces, and form three-dimensional extended frameworte. A number of theoretical studies have indicated that these hyperbolic structures, called "schwarzites" in honour of the mathematician Schwarz, should be more stable than the fullerenes [28]. In... 

So far, theoretically predicted schwarzites do not display this universality, although a number of predictions give an area per C atom close to that found in graphite and C60 (Fig. 2.22). These theoretical frameworks are the result of complex numerical quantum mechanical calculations. The apparent conservation of surface density, irrespective of the curvatures of the surface, is clearly not a direct consequence of standard physics. It will be very interesting to compare the surface densities of actual schwarzites (although they have yet to be prepared in the laboratory) with those of fullerenes and graphite. Given the usefulness of this principle in the study of tetrahedral frameworks, our bet is that they too will lie on the dotted line in Fig. 2.22. 

Figure 2.22 Area data calculated as described in Fig. 10 (using the same vertical sctde) for a range of hj perbolic "schwarzite" sp carbon networks predicted by v2uious theoreticians. The filled diamonds denote the corresponding areas per vertex for (phmar) graphite (n=6) and C o fullerene (n=5.62). (The latter is calculated from standard crystallographic data assuming a spherical network.)...

Klauda et al 21 use HM-IE ab initio calculations (see Section 3.1) to determine the effect of curvature of a carbon wall on its interactions with N2 and 02. Interactions with graphene, fullerene (Cgo) and schwarzite (Ci6s) are compared and curvature is seen to have a significant effect - particularly for the N2-carbon surface interactions. 

Harris [75] has also pointed out that his model based on fuUerene-hke elements has some connections with the so-called random schwarzite structure [94], which is based in turn on the ordered schwarzite structure [95] (the term schwarzite was coined after the German mathematician H.A. Schwarz, who first investigated the periodic minimal surfaces). The key feature of schwarzite is the occurrence... 

Figure 2.20 Possible structures for ordered schwarzite (a) and random schwarzite (b). (Reprinted with permission from Ref. [94]. 1992 The American Physical Society.)...

In connection with the latter point, we believe that the recent discovery [126] of a method to isolate individual graphenes may pave the way for unexpected findings in the field of carbons in general and in that of adsorption by carbons in particular. This first example of a truly bidimensional material that is also the thinnest conceivable object wiU shed light on whether or not graphenes of different sizes are flat or curved, continuous or discontinuous, and whether they may be made to curl into, e.g., random, or even ordered, schwarzites. We are undoubtedly at the dawn of an exciting new era of carbon science. 

Except for the fullerenes, carbon nanotubes, nanohoms, and schwarzites, porous carbons are usually disordered materials, and cannot at present be completely characterized experimentally. Methods such as X-ray and neutron scattering and high-resolution transmission electron microscopy (HRTEM) give partial structural information, but are not yet able to provide a complete description of the atomic structure. Nevertheless, atomistic models of carbons are needed in order to interpret experimental characterization data (adsorption isotherms, heats of adsorption, etc.). They are also a necessary ingredient of any theory or molecular simulation for the prediction of the behavior of adsorbed phases within carbons - including diffusion, adsorption, heat effects, phase transitions, and chemical reactivity. 

Regular porous carbons are carbon materials with a simple pore geometry they include carbon nanotubes, fullerenes, and schwarzites. If carbon nanotubes are considered for sensor applications, ab initio models are necessary to test whether adsorption of a certain molecule (e.g., NO2) generates a change of the electron density of the nanotube that is large enough to use the nanotube in a sensor [4]. 

Jiang, J., Wagner, N.J., and Sandler, S.I. (2004). A Monte Carlo simulation study of the effect of carbon topology on nitrogen adsorption on graphite, a nanotube bundle, C60 fullerite. Cl68 schwarzite, and a nanoporous carbon. Phys. Chem. Chem. Phys., 6, 4440-4. 

Next we will discuss a series of theoretical studies from Sandler and coworkers on fullerenes and schwarzites. Following a chronological sequence, we will start with adsorption on schwarzite, a hypothetical structure related to that of the... 

More recently, Jiang and Sandler carried out similar studies for the adsorption of CO2, N2, and their mixture [33] on the same schwarzite model adsorbent. As an illustration of their results. Fig. 14.2 shows the calculated (competitive) adsorption isotherms, as well as the selectivities of CO2 over N2 as a function of pressure for a CO2-N2 (0.21 0.79) mixture (the composition of this mixture corresponds to the flue gas emitted from the complete combustion of carbon with air). As the isotherms show, the use of the ab initio potential results in a larger difference between the amounts of adsorbed CO2 and N2... 

Figure 14.2 Left, adsorption isotherms of the COj-Nj mixture (hulk composition CO2/N2 = 0.21 0.79) in the Cj g schwarzite as a function of the total hulk pressure. Right, selectivity of CO2 over N2 as a function of the total hulk pressure (hulk composition CO2/N2 = 0.21 0.79) in the Cjgg schwarzite (with the Steele and ah initio potentials), sUicalite, Na-ZSM-5 (Si/Al = 23), and Na-ZSM-5 (Si/Al = 11). (Reprinted with permission from Ref. [33]. Copyright 2005 American Chemical Society.)...

Following their work with schwarzites, Sandler and coworkers carried out theoretical studies on N2 adsorption at 77 K on nd C7Q [36] using... 

Jiang, J. and Sandler, S.l. (2003). Monte Carlo simulation of O2 and N2 adsorption in nanoporous carbon (Cjgg schwarzite). Langmuir, 19, 3512-18. 

Babarao R, Hu Z, Jiang J et al (2007) Storage and separation of CO2 and CH4 in siU-cahte, C168 schwarzite, and lRMOF-1 a comparative study from monte carlo simulation. 

Other complex low-density carbon forms that are known to form including onions, nanotubes, foams and schwarzites [114, 115]. One interesting possible route to formation of carbon clathrates is the 3D-polymerization of fullerene samples, and one proposed 3D-form of a clathrate material (but not Type I or Type 11 materials described to date) has been observed experimentally [116]. 

FIGURE 35 Twofold symmetry axis views of the cubic D(c) and P(b) forms of schwarzite, emphasizing differences in the layout of the seven-member rings [35]. 

FIGURE 36 Views of two new crystalline schwarzites. Each has 216 carbon atoms per primitive unit cell with 80 six-member rings and 24 seven-member rings. The structure in (a) lies on a P minimal surface in a cubic cell 15.7 A on a side. The structure in (b) lies on a D minimal surface in an fee cell whose cubic lattice constant is 24.6 A [36]. 

As in previous topological modeling studies that have been recently devoted to grapheme layers with nanocones (Cataldo et al., 2010), Cgg fullerene stability (Vukicevic et al., 2011) or schwarzitic nanoribbons conformations (De Corato et al., 2012), the topological potential derives from the Wiener index W of the chemical structure (Estrada Hatano, 2010). 

Suzuki and coworkers [64] subsequently reported the synthesis and properties of the tetrabenzo[8]circulene 132, which can be seen as the repeating subunit of the hypothetical 3D curved graphene Schwarzite P192 III (Figure 5.18). 

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