### Chemical applications of topology and group theory. 29. Low density polymeric carbon allotropes based on negative curvature structuresChemical applications of topology and group theory. 29. Low density polymeric carbon allotropes based on negative curvature structures

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 Author King, R. B. Source United States Department of Energy Office of Scientific and Technical Information Content type Text Language English
 Subject Keyword MATERIALS SCIENCE ♦ MATHEMATICS, COMPUTERS, INFORMATION SCIENCE, MANAGEMENT, LAW, MISCELLANEOUS ♦ CARBON ♦ ALLOTROPY ♦ TOPOLOGY ♦ FULLERENES ♦ CRYSTAL STRUCTURE ♦ MOLECULAR STRUCTURE ♦ CALCULATION METHODS Abstract The construction of graphite and fullerene structures from networks of trigonal sp{sup 2} carbon atoms of zero and positive curvature can be extended to a fourth form of carbon, provisionally called schwarzites, consisting of networks of trigonal sp{sup 2} carbon atoms decorating infinite periodic minimal surfaces (IPMSs) of negative curvature and topological genus 3 such as the so-called D and P surfaces. The carbon networks of the simplest schwarzite structures contain only six- and seven-membered rings. The stable structures of both fullerenes and schwarzites contain enough six-membered rings so that no two five-membered rings in the fullerene structures or no two seven-membered rings in the schwarzite structures share edges leading to unstable pentalene and heptalene units, respectively. The smallest unit cell of a viable schwarzite structure of this type contains 168 carbon atoms and is constructed by applying a leapfrog transformation to a genus 3 figure containing 24 heptagons and 56 vertices. Although this C{sub 168} schwarzite unit cell has local O{sub h} point group symmetry based on the cubic lattice of the D or P surface, its larger permutational symmetry group is the PSL(2,7) group of order 168 analogous to the icosahedral pure rotation group, I, of order 60 of the C{sub 60} fullerene considered as the isomorphous PSL(2,5) group. The porosity of the IPMSs on which the schwarzite structures are based leads to predictions of unusually low density for this type of carbon allotrope. 38 refs., 11 figs., 5 tabs. ISSN 00223654 Educational Use Research Learning Resource Type Article Publisher Date 1996-09-12 Publisher Place United States Journal Journal of Physical Chemistry Volume Number 100 Issue Number 37