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We use inverse methods of statistical mechanics to explore trade-offsassociated with designing interactions to stabilize self-assembled structuresagainst changes in density or temperature. Specifically, we findisotropic,convex-repulsive pair potentials that maximize the density range forwhich a two-dimensional square lattice is the stable ground state subject to aconstraint on the chemical potential advantage it exhibits over competingstructures (i.e., 'depth' of the associated minimum on the chemical potentialhypersurface). We formulate the design problem as a non-linear program, whichwe solve numerically. This allows us to efficiently find optimized interactionsfor a wide range of possible chemical potential constraints. We find thatassemblies designed to exhibit a large chemical potential advantage at aspecified density have a smaller overall range of densities for which they arestable. This trend can be understood by considering the separation-dependentfeatures of the pair potential and its gradient required to enhance thestability of the target structure relative to competitors. Using moleculardynamics simulations, we further show that potentials designed with largerchemical potential advantages exhibit higher melting temperatures.

Breadth versus depth: Interactions that stabilize particle assemblies to changes in density or temperature