Using Escher's Genetic Code To Program Self-Assembling Colloidal Films

The properties of many two-dimensional (2D) colloidal materials rely on the precise spatial organization of their constituents. Colloids are considered facile building blocks whose shape, size, and surface functionality can be modified, enabling self-assembly into a wide range of patterns. Notwithstanding the experimental challenges of realizing some designs, a major conceptual challenge is understanding the connection between the final structures and optimal sets of colloidal features to achieve them. In silico studies can elucidate and test these relationships. Here, we develop a simple method for deriving surface functionalization patterns from isohedral tilings that induce self-assembly into any chosen 2D symmetry group at a planar interface. The result is a sequence of letters, s ∈ {A, T, C, G}, or a gene, that describes the colloid’s anisotropic shape and chemical patterning. This topological genome is finite and can be exhaustively enumerated. Moreover, these genes are human readable and can be used to intuitively design colloids. This technique is inspired by some of Maurits Cornelis (M. C.) Escher’s methods and the so-called “Escherization problem.” We illustrate how to optimize features of a gene to design patchy particles that self-assemble into a film with chosen symmetry and porosity determined a priori.