Novel Self-Assembled Morphologies from Isotropic Interactions

Abstract

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...

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