From Kelvin problem to Kelvin carbons.

As children's toys, soap bubbles also underpin many important scientific questions. What is the most efficient structure for foam? - the "Kelvin problem," has been one of the most intriguing science and widely discussed over the past hundred years. Soap bubbles' frameworks have similar topology with sp(3)-bonded carbon or silicon allotropes, e.g., Weaire-Phelan foam and superconducting clathrate Na8Si46. By looking at the most efficient structure for foams, we construct a series of new carbon allotropes, named "Kelvin carbons." Unexpectedly, all 11 Kelvin carbons are structurally stable wide-bandgap semiconductors, and their densities and their intrinsic hardnesses are both about 81%-87% that of diamond. The seventh state of Kelvin carbons, K(VII), is a very low energy carbon structure after graphite, diamond, lonsdaleite, and type-II carbon clathrate. Kelvin carbons, which constitute a "bridge" between macro-foams and micro-carbons, together with recently proposed novel carbon phases can enrich the study of carbon allotropes.