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.

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.

Patchy colloids are promising candidates for building blocks in directed self-assembly, but large scale synthesis of colloids with controlled surface patterns remains challenging. One potential fabrication method is to self-assemble the surface patterns themselves, allowing complex morphologies to organize spontaneously. For this approach to be competitive, prediction and control of the pattern formation process are necessary. However, structure formation in many-body systems is fundamentally hard to understand, and new theoretical methods are needed. Here we present a theory for self-assembling pattern formation in multi-component systems on the surfaces of colloidal particles, formulated as an analytic technique that predicts morphologies directly from the interactions in an effective model. As a demonstration we formulate an isotropic model of alkanethiols on gold, a suggested system for directed self-assembly, and predict its morphologies and transitions as a function of the interaction parameters.

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.PACS numbers: 89.75. Kd, 75.10.Hk, 05.65.+b Understanding the principles behind spontaneous formation of structured morphologies is of interest both as a fundamental scientific question and in engineering applications where the possibility of using self-assembly to produce novel materials provides a compelling complement to traditional blue-printed fabrication [1,2]. Consequently there is growing interest in exploring interactions that can facilitate self-fabrication of materials with novel properties [3][4][5]. The more general question of what structures are possible to self-assemble from a given class of interactions has however not been addressed, with some notable exceptions e.g. simulation based analysis of polyhedral packing [6].To examine the possibilities of self-assembly from a theoretical angle, in this Letter we consider systems with pair-wise isotropic interactions, frequently used as coarse grained models of more complex meso-scale systems such as colloidal systems [7], particles in an ambient fluid [8,9], and spin glasses [10]. We show that typical aggregates appearing as low temperature configurations in particle simulations with randomly generated medium range isotropic interactions can be predicted analytically using an adaptation of the spherical spin model [11]. The morphologies, many of them novel and surprisingly complex, can be systematically classified by their spontaneous breaking of the rotational symmetry.To formulate a solvable model of a particle system we start by considering a lattice spin system with Hamiltonian on the formwhere s i ∈ {0, 1}, and s i = 1 represents a particle at lattice site i and s i = 0 represents vacuum. The total number of particles is set by the normalization i s i = N . The interaction U ij is an effective isotropic potential with a hard core repulsion corresponding to the lattice spacing. In a spin glass metal the interaction is mediated by a polarization of the Fermi sea [12] while in a colloidal system it could involve a surface polymer induced steric hindrance competing with a depletion attraction [13]. The discrete model described by Eq. (1) can equivalently be formulated as a continuous one (s i ∈ R) with auxiliary constraints, i s m i = N ∀m. To make the model analytically tractable we relax the constraints to include only the first two moments. The result is the spherical spin model including an external field, with Hamiltonian H s = ij U ij s i s j + h i s i . A necessary condition for an energy minimum in this model is j (U ij − λδ ij )s j = h/2, where δ is the identity...

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.