Frustrated Self-Assembly of Non-Euclidean Crystals of Nanoparticles*

Many complex structures in nature such as living organisms self-organize from simple units. They display high complexity, which allows multiple functions, and at the same time are efficient, adaptable and robust.
Can we program the self-assembly of three-dimensional complex structures with simple building blocks, and reach similar or higher level of sophistication in engineered materials?
We present an analytic theory of tetrahedral nanoparticles self-assembling in 3D space, where unavoidable geometrical frustration combined with competing attractive and repulsive inter-particle interactions lead to controllable, high-yield, and enantiopure self-assembly of quasi-2D helicoidal ribbons. This theory, based on crystal structures in non-Euclidean space, predicts morphologies that exhibit qualitative agreement with experimental observations. We expect that this theory may be generalizable to the self-assembly of simple polyhedral building blocks into complex morphologies with new material capabilities such as tunable optical activity, essential for multiple emerging technologies.