Using Symmetry to Direct Two-dimensional Colloidal Crystal Self-assembly

We investigate self-assembling rings that can template the organization of an arbitrary colloidal unit into any desired periodic symmetry at a planar interface. By viewing this as a tiling problem, we illustrate how the shape and chemical functionality of these rings may be derived from symmetry, rather than through iterative computational methods such as inverse design. Moreover, we illustrate how these features are reflected by their orbifold symbol, which provides a natural language to express them. We performed molecular dynamics simulations to observe the self-assembly of these rings and found 5 different characteristics which could be easily rationalized based on their orbifold. These include systems which (1) undergo chiral phase separation, (2) are addressably complex, (3) exhibit self-limiting growth into clusters, (4) may form a smectic phase, and (5) those from symmetry groups which allow one to select rings which exhibit different self-assembly behaviors. We discuss how a ring's curvature plays an integral role in achieving correct self-assembly, in practice, and we illustrate how to derive these ring shapes. This provides a method for patterning colloidal systems at interfaces without explicitly programming this information onto the colloidal unit itself.