Tetrahedral Colloidal Clusters from Random Parking of Bidisperse Spheres

Using experiments and simulations, we investigate the clusters that form when colloidal spheres stick irreversibly to—or “park” on—smaller spheres. We use either oppositely charged particles or particles labeled with complementary DNA sequences, and we vary the ratio $\alpha$ of large to small sphere radii. Once bound, the large spheres cannot rearrange, and thus the clusters do not form dense or symmetric packings. Nevertheless, this stochastic aggregation process yields a remarkably narrow distribution of clusters with nearly 90% tetrahedra at $\alpha =2.45$. The high yield of tetrahedra, which reaches 100% in simulations at $\alpha =2.41$, arises not simply because of packing constraints, but also because of the existence of a long-time lower bound that we call the “minimum parking” number. We derive this lower bound from solutions to the classic mathematical problem of spherical covering, and we show that there is a critical size ratio ${\alpha }_c=(1+\sqrt{2})\approx 2.41$, close to the observed point of maximum yield, where the lower bound equals the upper bound set by packing constraints. The emergence of a critical value in a random aggregation process offers a robust method to assemble uniform clusters for a variety of applications, including metamaterials.

Figure 1

(a) Two colloidal sphere species are mixed together to form clusters. (b) Oppositely charged polystyrene spheres cluster due to electrostatic attraction. Optical micrograph shows a tetramer ($N=4$). (c) Polystyrene spheres labeled with complementary DNA strands (not to scale) cluster due to DNA hybridization. Optical micrograph shows a trimer ($N=3$); the small, central sphere is fluorescent.

Figure 2

(a) Yield curves, as determined by simulations, for $N$-particle clusters, $2\le N\le 8$, where the critical size ratio ${\alpha }_c$ is marked with a black line. Below are histograms for (b) DNA-labeled particles (left) at $\alpha =1.90$ and (c) charged particles (right) at $\alpha =2.45$, as observed in experiments (colored bars) and as predicted from simulations (gray bars). Error bars are 95% confidence intervals (Wilson score interval method).

Figure 3

$N_{\max }$ (solid gray) and $N_{\min }$ (black) as functions of $\alpha$. Cluster images show sphere configurations at discontinuities of these curves. Average cluster sizes from simulations (dashed gray line) and experiments (blue and red data points) are shown. We characterize the statistical dispersion in each distribution by the average absolute deviation from the median, indicated by dotted light gray lines for simulations and vertical bars for experiments.