Designing the ground state is not enough: Lessons from the self-assembly of Archimedean shells

Designing particle interactions such that a target structure is the thermodynamic ground state is a central paradigm in self-assembly. However, ensuring that the target is lowest in energy and that obvious competitors are energetically penalized does not, by itself, guarantee successful assembly at finite temperature, since even under these conditions competing structures can reappear as minima of the free-energy landscape. Focusing on the colloidal Archimedean snub-cube, we compute the full free-energy landscape of all competing aggregates using a cluster-based thermodynamic approach. While the target structure is uniquely selected at the level of potential energy, we find that competing clusters can become thermodynamically favored due to entropic contributions, particularly when bond directionality is high. In this regime, incomplete structures, such as icosahedra, are stabilized despite their higher energy per particle, leading to a dramatic suppression of the target yield. Our results provide quantitative guidelines for inverse design strategies that explicitly account for entropy, bond flexibility, and experimental conditions.