Aggregation Transitions in a Minimal Model of Geometrically Frustrated Assembly

Geometric frustration has been recognized as an important framework for understanding a wide range of self-assembling systems where the balance between the cohesive gains of assembly and the cost of frustration can result in unique, scale-dependent thermodynamics. One salient behavior is the existence of equilibrium states characterized by finite-sized aggregates that are much larger in size than the individual sub-units they are made of. The continuum models used to describe these assemblies derive their thermodynamics from the ground-state elastic energetics - ignoring finite temperature effects. We describe a 2D lattice model of geometrically frustrated assembly, which encodes both positional and “shape misfit” degrees of freedom of assembling subunits. We exploit this model to examine the relationship between aggregation temperature, concentration and frustration. In the continuum limit, we find that - for fixed sub-unit concentration - increasing frustration depresses the critical aggregation temperature, while also reducing the mean aggregate dimensions. These predictions are explored via Monte Carlo simulation. This work serves as a first step towards a more detailed understanding of the thermodynamics of geometrically frustrated assembly.