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.

Figure 1

Predicted morphological alphabet for aggregating potentials generated by Eq. (3) (with $\omega \le 8$). The morphologies shown are energetically preferable for a large fraction of random potentials. Red background indicates a strong signal while blue background signifies weaker signal. $\sigma$ denotes the number of active radial oscillations of the ground frequency after the threshold mapping.

Figure 2

Self-assembled morphologies for a single potential. (a) A typical energy spectrum of a random potential. The wavelength of the morphologies is determined by the interior minimum $\kappa$ of the spectrum when the number of particles $N$ is less than $N_c$, the number required to build a disk with radius corresponding to $k_0$; see Eq. (4). (b)–(g) Transitions between different symmetry groups for the potential of (a) as the particle number is decreased. (Top) Configurations from annealed particle simulations, (middle) corresponding states of the spherical model with contours added at a level giving correct number of particles, and (bottom) Bessel spectra (arb. units) of the particle configurations expanded on the form of (3), summed over $\omega$. The first maximum corresponds to the mass builder $k_m$, the second should be compared to $\kappa =1$ predicted from the spectrum in (a), and the rest are overtones.

Figure 3

Examples of configurations from particle simulations with randomly generated potentials. All configurations except (b) are well represented by functions on the form (3) with the first four terms included, most requiring only one or two. (a) Morphologies predicted as common by our method; see Fig. 1. (b) An example of a hierarchical structure, not directly describable in our theory, where each part of the usual Bessel structure serves as the nucleation point for a separate Bessel structure. (c) Morphologies that, while being simply representable by Eq. (3), are not predicted in Fig. 1.

Figure 4

Decreasing the number of particles causes a transition from disklike structures at the macroscale to more complex morphologies at the mesoscale. The figure shows the phase diagram with the predicted (full line) and measured (markers) critical number of particles $N_c$, adjusted for packing fraction $f$ versus $k_0$ calculated [see Fig. 2a] from respective spectra for 33 random potentials. Inset: Four examples of potentials used.