Tetrahedral colloidal clusters from random parking of bidisperse spheres

TL;DR

This study combines experiments and simulations to analyze how random parking of bidisperse spheres leads to predominantly tetrahedral colloidal clusters, revealing a critical size ratio that maximizes tetrahedron formation.

Contribution

It introduces a novel theoretical framework linking spherical covering solutions to colloidal cluster formation, identifying a critical size ratio for optimal tetrahedral assembly.

Findings

Nearly 90% tetrahedra at specific size ratio in experiments

100% tetrahedra yield in simulations at a close size ratio

Identification of a critical size ratio (~2.41) for maximum tetrahedral clusters

Abstract

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.