Erik Edlund

Universality of striped morphologies.

We present a method for predicting the low-temperature behavior of spherical and Ising spin models with isotropic potentials. For the spherical model the characteristic length scales of the ground states are exactly determined but the morphology is shown to be degenerate with checkerboard patterns, stripes and more complex morphologies having identical energy. For the Ising models we show that the discretization breaks the degeneracy causing striped morphologies to be energetically favored and therefore they arise universally as ground states to potentials whose Hankel transforms have nontrivial minima.

Designing Isotropic Interactions for Self-Assembly of Complex Lattices

We present a direct method for solving the inverse problem of designing isotropic potentials that cause self-assembly into target lattices. Each potential is constructed by matching its energy spectrum to the reciprocal representation of the lattice to guarantee that the desired structure is a ground state. We use the method to self-assemble complex lattices not previously achieved with isotropic potentials, such as a snub square tiling and the kagome lattice. The latter is especially interesting because it provides the crucial geometric frustration in several proposed spin liquids.

Novel Self-Assembled Morphologies from Isotropic Interactions

We present results from particle simulations with isotropic medium range interactions in two dimensions. At low temperature novel types of aggregated structures appear. We show that these structures can be explained by spontaneous symmetry breaking in analytic solutions to an adaptation of the spherical spin model. We predict the critical particle number where the symmetry breaking occurs and show that the resulting phase diagram agrees well with results from particle simulations.

Using the uncertainty principle to design simple interactions for targeted self-assembly

We present a method that systematically simplifies isotropic interactions designed for targeted self-assembly. The uncertainty principle is used to show that an optimal simplification is achieved by a combination of heat kernel smoothing and Gaussian screening of the interaction potential in real and reciprocal space. We use this method to analytically design isotropic interactions for self-assembly of complex lattices and of materials with functional properties. The derived interactions are simple enough to narrow the gap between theory and experimental implementation of theory based designed self-assembling materials.

Chiral Surfaces Self-Assembling in One-Component Systems with Isotropic Interactions

We show that chiral symmetry can be broken spontaneously in one-component systems with isotropic interactions, i.e., many-particle systems having maximal a priori symmetry. This is achieved by designing isotropic potentials that lead to self-assembly of chiral surfaces. We demonstrate the principle on a simple chiral lattice and on a more complex lattice with chiral supercells. In addition, we show that the complex lattice has interesting melting behavior with multiple morphologically distinct phases that we argue can be qualitatively predicted from the design of the interaction.

Renormalization of Cellular Automata and Self-Similarity

We study self-similarity in one-dimensional probabilistic cellular automata (PCA) using the renormalization technique. We introduce a general framework for algebraic construction of renormalization groups (RG) on cellular automata and apply it to exhaustively search the rule space for automata displaying dynamic criticality.Previous studies have shown that there exists several exactly renormalizable deterministic automata. We show that the RG fixed points for such self-similar CA are unstable in all directions under renormalization. This implies that the large scale structure of self-similar deterministic elementary cellular automata is destroyed by any finite error probability.As a second result we show that the only non-trivial critical PCA are the different versions of the well-studied phenomenon of directed percolation. We discuss how the second result supports a conjecture regarding the universality class for dynamic criticality defined by directed percolation.

A design path for the hierarchical self-assembly of patchy colloidal particles

Patchy colloidal particles are promising candidates for building blocks in directed self-assembly. To be successful the surface patterns need to be simple enough to be synthesized, while feature-rich enough to cause the colloidal particles to self-assemble into desired structures. Achieving this is a challenge for traditional synthesis methods. Recently it has been suggested that surface patterns themselves can be made to self-assemble. In this paper we present a design path for the hierarchical targeted self-assembly of patchy colloidal particles based on self-assembling surface patterns. At the level of the surface structure, we use a predictive method utilizing the universality of stripes and spots, coupled with stoichiometric constraints, to cause highly specific and functional patterns to self-assemble on spherical surfaces. We use a minimalistic model of an alkanethiol on gold as a demonstration, showing that even with limited control over the interaction between surface constituents we can obtain patterns that cause the colloidal particles themselves to self-assemble into various complex geometric structures, such as strings, membranes, cubic aggregates and colloidosomes, as well as various crystalline patterns.