Thermal stability and secondary aggregation in geometrically-frustrated assembly

Geometric frustration has emerged as a paradigm for potentially engineering self-limiting structures with finite, well-controlled sizes much larger than a single subunit. Frustrated assemblies often escape to unlimited sizes through elastic shape deformations or through the inclusion of defects or broken bonds within the assembled structures. Understanding how to prevent these modes of escape is crucial to controlling assembly size. While the zero-temperature mechanics of geometrically-frustrated assemblies is well-studied, less is known about the role of temperature. In the absence of thermal fluctuations, any partial bonding between subunits, regardless of how weak, will always lower the energy of the system, resulting in an unlimited aggregate of weakly-bonded structures. Thus, thermal fluctuations must play an important role in the stability of self-limiting structures. We here consider a simple model of a frustrated 1D incommensurate chain at finite temperature. We show that a quantity termed the "defectability", which characterizes the tendency for defect formation, determines whether there exists a range of subunit concentrations and temperatures in which self-limiting assembly can occur. In particularly, we find that there is a frustration-dependent minimum temperature required for self-limiting assembly. Low-temperature condensation of finite-aggregates is likely the generic result of weak-binding possible in any physical frustrated assembly. Hence, our analysis highlights key principles needed for targeting, designing and stabilizing experimental systems that regain robust self-limiting regimes.