Launching Graphene into 3D-Space: Symmetry, Topology, and Strategies for Bottom-Up Synthesis of Schwarzites

Schwarzites are hypothetical carbon allotropes in the form of a continuous negatively curved surface with a three-dimensional periodicity. These materials of the future attract interest because of the expectation for their large surface area per volume, high porosity, tunable electric conductivity, and excellent mechanical strength combined with light weight. Three-decades long history attempting synthesize Schwarzites from the gas phase carbon atoms went without success. Design of Schwarzites is both digital art and science of placing tiles of sp2-carbon polygons on mathematically defined triply periodic minimal surfaces. The knowledge of how to connect polygons in sequence using the rules of symmetry unlocks paths for the bottom-up synthesis of Schwarzites by organic chemistry methods. Schwarzite tiling by heptagons is systematically analyzed and classified by symmetry and topology. For the first time, complete plans for a bottom-up synthesis of many Schwarzites are demonstrated. A trimer of heptagons is suggested as the key building block for most synthetic schemes.