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Geometric frustration is recognized to generate complex morphologies in self-assembling particulate and molecular systems. In bulk states, frustration drives structured arrays of topological defects. In the dilute limit, these systems have been shown to form a novel state of self-limiting assembly, in which the equilibrium size of multiparticle domains are finite and well defined. In this article we employ Monte Carlo simulations of a recently developed 2D lattice model of geometrically frustrated assembly [Hackney et al., Phys. Rev. X 13, 041010 (2023)] to study the phase transitions between the self-limiting and defect bulk phase driven by two distinct mechanisms: (1) increasing concentration and (2) decreasing temperature or frustration. The first transition is mediated by a concentration-driven percolation transition of self-limiting, wormlike domains into an intermediate heterogeneous network mesophase, which gradually fills in at high concentration to form a quasiuniform defect bulk state. We find that the percolation threshold is weakly dependent on frustration and shifts to higher concentration as frustration is increased, but depends strongly on the ratio of cohesion to elastic stiffness in the model. The second transition takes place between self-limiting assembly at high-temperature or frustration and phase separation into a condensed bulk state at low temperature or frustration. We consider the competing influences that translational and conformational entropy have on the critical temperature or frustration and show that the self-limiting phase is stabilized at higher frustrations and temperatures than previously expected. Taken together, this understanding of the transition pathways from self-limiting to bulk defect phases of frustrated assembly allows us to map the phase behavior of this 2D minimal model over the full range of concentration. 

In self-assembling systems, geometric frustration leads to complex states characterized by internal gradients of shape misfit. Frustrated assemblies have drawn recent interest due to the unique possibility that their thermodynamics can sense and select the finite size of assembly at length scales much larger than constituent building blocks or their interactions. At present, self-limitation is chiefly understood to derive from zero-temperature considerations, specifically the competition between cohesion and scale-dependent elastic costs of frustration. While effects of entropy and finite-temperature fluctuations are necessarily significant for self-assembling systems, their impact on the self-limiting states of frustrated assemblies is not known. We introduce a generic, minimal model of frustrated assembly and establish its finite-temperature and concentration-dependent thermodynamics by way of simulation and continuum theory. The phase diagram is marked by three distinct states of translation order: a dispersed vapor, a defect-riddled condensate, and the self-limiting aggregate state. We show that, at finite temperature, the self-limiting state is stable at intermediate frustration. Furthermore, in contrast to the prevailing picture, its thermodynamic boundaries with the macroscopic disperse and bulk states are temperature controlled, pointing to the essential importance of translational and conformational entropy in their formation.