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.

Figure 1

The kagome lattice in real space (a) as described by its first few reciprocal lattice vectors (b). Structure factors for the included reciprocal vectors are shown in black and primitive lattice vectors ${\mathbf{k}}_p$ as gray arrows. The kagome lattice (illustrated in black, with basis and lattice vectors in white) is indeed determined by this limited part of the reciprocal lattice.

Figure 2

A spectral method for targeted self-assembly. By designing an energy spectrum $\widehat{V}(k)$ (a) with minima at the peaks of $\widehat{\rho }$ and taking its Fourier transform we arrive at a potential $V(r)$ (b) that causes particles to self-assemble into a target lattice. (c) Time evolution of a self-assembling kagome lattice. The number of grains diminishes by random walk of their boundaries, a process faster for smaller systems (d) but where some local defects are still present as the dimensions of the system is not fine-tuned to fit the lattice.

Figure 3

Isotropic potentials that cause particles to self-assemble into snub square (a) and honeycomb (b) lattices.