Spectral Methods for Self-Assembly

Abstract: Conventional manufacturing relies on robotics and chemistry to create, shape and combine components into products. Nature shows us an alternative route: by carefully designing interactions between components, they can be made to spontaneously self-assemble into complex arrangements. The field of self-assembly has seen a series of experimental triumphs, but there is a relative dearth of theoretical results of the kind expected in physics. This thesis present some analytical results and their connection to self-assembly. The common theme is a focus on the energy spectrum of the (isotropic) interactions of a given model. Paper I solves an adaptation of the spherical spin model, demonstrating the centrality of the energy spectrum and predicting universal striped behavior. Paper II derives an alternative set of solutions, relevant for aggregating particle systems, and predicts a large but limited morphological alphabet. They both describe what happens typically, in systems with general interactions. Paper III represents a different take on the spectral picture, not trying to understand general interactions but to design specific ones. It presents a method that allows us to self-assemble complex crystal structures by matching an energy spectrum to the Fourier spectrum of the target structure. Interestingly, the method produces interactions for which we can prove that the target structure is a ground state.