Symmetry-based discovery of multicomponent, two-dimensional colloidal crystals

We present a systematic method for computing the ground state phase behavior of multicomponent colloidal materials. In two-dimensions there are exactly 17 “wallpaper groups” which represent distinct combinations of isometries of the Euclidean plane. Using properties of these groups, we develop an algorithm to cover the plane with a fixed number of arbitrary components in all ways that satisfy a desired stoichiometric ratio. These combined symmetry-stoichiometry rules dramatically reduce the number of possible configurations, which generally suffer from a so-called “combinatorial explosion” otherwise making extensive, random structure searching computationally infeasible. With subsequent continuum relaxation, this enumeration approach can predict crystal structures in silico for multicomponent colloidal mixtures. We use this approach to investigate the ground state phase behavior of multicomponent systems inspired by DNA-coated colloidal mixtures, with a focus on stable, low-density “open” crystals. We demonstrate the approach for binary and ternary mixtures at zero ambient pressure to explore how complexity can be achieved through the combination of several components with simple interactions rather than a single component with a more complicated interaction potential.