Frustrated self-stacking assembly of non-Euclidean shell colloids

When geometrically frustrated particles self-assemble local inter-subunit shape misfits can propagate to large scale strain gradients that give rise to finitely sized, self-limiting structures in equilibrium. Recently, a model of deformable, cylindrically curved, colloidal shells investigated the roles of bending and inter-particle bond stretching energies within one-dimensional particle stacks and established a transition from self-limiting to unlimited assembly behavior that depends on the range of attraction and particle geometry. However, shape deformations of shells with curvature in two directions (Gaussian curvature) often involve stretching which incurs elastic penalties that are typically much greater than bending. Here, we propose a model of elastic, doubly curved, shallow shells and investigate the roles of Gaussian curvature and stretching energy in curvature-frustrated shell stacks and the consequences for self-limiting behavior and size selection. Through a combination of continuum elastic theory and coarse-grained simulations we identify key particle variables that control when stretching dominates over bending and uncover the regimes of self-limiting behavior for saddle, spherical and cylindrically shaped particles. These predictions provide critical design criteria for experimental realizations of self-limiting frustrated particle systems including controlled assemblies of polymeric microshells or curved DNA origami nanoparticles.