The first space-filling polyhedrons of polymer cubic cells originated from Weaire-Phelan structure created by polymerization induced phase separation

Abstract

The Weaire–Phelan structure is a three-dimensional structure composed of two different polyhedra having the same volume, i.e., pyritohedron and truncated hexagonal trapezohedron. It was proposed by Weaire and Phelan in 1993 as a solution of the Kelvin problem of filling space with no gaps with cells of minimum surface area and equal volume. It was found in physical systems including liquid foam and a metal alloy while it has never been constructed as organic materials. We report herewith the first polymeric Weaire–Phelan structure constructed through phase-separation of a single polymer species that is synthesized by simple polyaddition between tetrakis(3-mercaptopropionate) and 1,6-diisocyanatohexane. The structure has the order of micrometers and is amorphous unlike reported crystal structures similar to the Weaire–Phelan structure.