Controlled Self-Assembly of Periodic and Aperiodic Cluster Crystals

Soft particles are known to overlap and form stable clusters that self-assemble into periodic crystalline phases with density-independent lattice constants. We use molecular dynamics simulations in two dimensions to demonstrate that, through a judicious design of an isotropic pair potential, one can control the ordering of the clusters and generate a variety of phases, including decagonal and dodecagonal quasicrystals. Our results confirm analytical predictions based on a mean-field approximation, providing insight into the stabilization of quasicrystals in soft macromolecular systems, and suggesting a practical approach for their controlled self-assembly in laboratory realizations using synthesized soft-matter particles.

Figure 1

Pair potentials used in this study. (a) Real space $U(r)$, normalized such that $U(0)=1$. (b) Fourier transform $\tilde{U}(k)$, where the first minimum is always at $k=1$, and the second minimum is located with increasing order at $k_4=\sqrt{2}$ (solid blue), $k_5=(1+\sqrt{5})/2$ (dot-dashed red), $k_6=\sqrt{3}$ (dashed brown), $k_{12}=\sqrt{2+\sqrt{3}}$ (dotted green), and $k_{\infty }=2$ (double-dot-dashed cyan). The inset shows a close-up view of the minima.

Figure 2

Self-assembled cluster crystals for the pair potentials in Fig. 1. We generate (a) stripes, periodic (b) tetragonal, and (c) hexagonal, as well as quasiperiodic (d) decagonal and (e) dodecagonal cluster crystals. Simulation parameters are $N=16384$, $T=0.03$, and $\overline{c}=0.8$ (a),(e), $\overline{c}=0.9$ (b), $\overline{c}=0.7$ (c), $\overline{c}=0.6$ (d). The top, middle, and bottom rows show snapshots in real space comparing MD results (red circles) with mean-field predictions for $c(\mathit{r})$ (in gray scale at the bottom-right corners), diffraction diagrams, and radial distribution functions. Particles are drawn with radius 1. The real-space view is limited to about 20% of the simulation box [27].

Figure 3

Coexistence of the liquid and cluster crystal phases, shown as bars, as a function of mean particle density $\overline{c}$ from simulation data. The red line is the mean-field spinodal (sp) instability limit. Cluster crystals with $n$-fold rotational symmetry are observed exclusively for $\overline{c}$ above the vertical dashed lines, while hexagonal cluster crystals are found below. Densities are slightly shifted horizontally by $\pm 0.01$ among different cluster crystals for better visibility.

Figure 4

Histograms of the cluster sizes in Fig. 2. (a),(b) Periodic crystals show sharp distributions. (c),(d) Quasicrystals exhibit broad distributions. We use the cluster size cutoff parameter $\text{MinPts}=8$ for the DBSCAN algorithm [31].

Figure 5

(a) At low temperature and high density the dodecagonal quasicrystal can reversibly transform into a compressed hexagonal phase [27]. (b) Snapshot of the liquid phase at a temperature just above the onset of ordering, possibly showing a liquid of clusters. Clusters are colored according to their size from small (light blue, size $\le 10$) to large (dark red, size $\ge 30$).