Self-limited stacking of non-Euclidean colloidal shells: From saddles to caps

Abstract

The presence of geometric frustration in self-assemblies, stemming from a shape misfit between identical subunits, can lead to the spontaneous formation of finite-sized structures known as self-limitation. Recently proposed models of curved, colloidal shells, dubbed "curvamers", that form one-dimensional stacks have shown curvature-induced frustration to be a promising class of particle designs for engineering self-limiting assembly. However, existing models of curvamer assembly considered only cylindrically curved shells where elastic costs are derived purely from bending. Here, we study the self-limiting behavior of a more general class of curvature-frustrated colloidal shells, extending shapes to include non-Euclidean geometries such as spherical caps and saddles whose deformations can also incur stretching costs. We introduce continuum mechanical and discrete, coarse-grained models of shell stacks to study how qualitatively different intra-assembly strain gradients required for each geometry affect the buildup of super-extensive elastic costs. We find the accumulation of elastic energies in non-Euclidean shell stacks is sensitive to shell thickness via a dimensionless quantity related to the Foppl-von Karman number that characterizes the relative strength of stretching to bending. In particular, thin shells penalize stretching more causing faster elastic energy growth resulting in the suppression of self-limitation with sizes generally largest for cylindrical shells, smallest for spherical caps and intermediate for saddles. We additionally find that while thin spherical caps experience the strongest elastic penalties, the rapidly increasing rate of elastic penalties in stacks of saddles lead to a robust size-selection as inter-particle binding strength grows as well as relatively smaller dispersity in self-limiting stack size in comparison to cylindrical and spherical curvamers.