Pentaheptite Modifications of the Graphite Sheet

Abstract

Pentaheptites (three-coordinate tilings of the plane by pentagons and heptagons only) are classified under the chemically motivated restriction that all pentagons occur in isolated pairs and all heptagons have three heptagonal neighbors. They span a continuum between the two lattices exemplified by the boron nets in ThMoB4 (cmm) and YCrB4 (pgg), in analogy with the crossover from cubic-close-packed to hexagonal-close-packed packings in 3D. Symmetries realizable for these pentaheptite layers are three strip groups (periodic in one dimension), p1a1, p112, and p111, and five Fedorov groups (periodic in two dimensions), cmm, pgg, pg, p2, and p1. All can be constructed by simultaneous rotation of the central bonds of pyrene tilings of the graphite sheet. The unique lattice of cmm symmetry corresponds to the previously proposed pentaheptite carbon metal. Analogous pentagon-heptagon tilings on other surfaces including the torus, Klein bottle, and cylinder, face-regular tilings of pentagons and b-gons, and a full characterization of tilings involving isolated pairs and/or triples of pentagons are presented. The Kelvin paradigm of a continuum of structures arising from propagation of two original motifs has many potential applications in 2D and 3D.

Penta-graphene: A new carbon allotrope

“…Total energy calculations are performed to investigate the thermodynamic stability of penta-graphene. Although this phase is metastable compared with graphene and previously reported 2D carbon allotropes (6,11,14) due to its violation of the IPR, it is more stable than some nanoporous carbon phases such as 3D T-carbon (25), 2D α-graphyne (6), and (3, 12)-carbon sheet (26) (Fig. 2A).…”

“…6C, confirming its dynamical stability. Indeed, more complicated structures are expected to be built, akin to the pentaheptite modification of graphene (14). An example of such structures containing four penta-graphene layers (24 atoms) per unit cell, termed ABAA-T24, is presented in SI Appendix, Fig.…”

“…For instance, C 60 consists of 12 pentagons separated by 20 hexagons forming the shape of a soccer ball, which is a perfect footnote to IPR. The emergence of carbon pentagons is also found to be accompanied by carbon heptagons in some cases (14), but are separated from each other. Inspired by the synthesis of pure pentagon-based C 20 cage (16), considerable effort has been made to stabilize fusedpentagon-based and non-IPR carbon materials in various dimensionalities (10,15).…”

“…Extended carbon networks composed of only pentagons are rarely seen. Carbon pentagons are usually considered as topological defects or geometrical frustrations (14) as stated in the well-known "isolated pentagon rule" (IPR) (15) for fullerenes, where pentagons must be separated from each other by surrounding hexagons to reduce the steric stress. For instance, C 60 consists of 12 pentagons separated by 20 hexagons forming the shape of a soccer ball, which is a perfect footnote to IPR.…”

First principles calculations of pentaheptite graphene and boronitrene derivatives

“…This implies that pentaheptites are generated by simultaneous creation of SW-type defects over the entire graphene sheet, transforming pyrenelike rings into symmetrically paired pentagons and heptagons. 37,49 Although there are various ways in which the R 5 and R 7 rings can be constructed, it has been established that pentaheptites also adhere to the rule that the most favorable configurations are those which minimize the number of pentagon-pentagon pairs and maximize the number of pentagon-heptagon pairs, as is the case with fullerenes and haeckelites (cf. Refs.…”