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Using experiments and simulations, we investigate the clusters that form whencolloidal spheres stick irreversibly to -- or "park" on -- smaller spheres. Weuse either oppositely charged particles or particles labeled with complementaryDNA 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 notform dense or symmetric packings. Nevertheless, this stochastic aggregationprocess 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 packingconstraints, but also because of the existence of a long-time lower bound thatwe call the "minimum parking" number. We derive this lower bound from solutionsto the classic mathematical problem of spherical covering, and we show thatthere is a critical size ratio $\alpha_c=(1+\sqrt{2})\approx 2.41$, close tothe observed point of maximum yield, where the lower bound equals the upperbound set by packing constraints. The emergence of a critical value in a randomaggregation process offers a robust method to assemble uniform clusters for avariety of applications, including metamaterials.

Tetrahedral colloidal clusters from random parking of bidisperse spheres