  Many recent studies of liquid-liquid phase separation in biology focus on
phase separation as a dynamic control mechanism for cellular function, but it
can also result in complex mesoscopic structures. We primarily investigate a
model system consisting of DNA nanostars: finite-valence, self-assembled
particles that form micron-scale liquid droplets via a binodal phase
transition. We demonstrate that, upon phase separation, nanostar droplets
spontaneously form hyperuniform structures, a type of disordered material with
`hidden order' that combines the long-range order of crystals with the
short-range isotropy of liquids. We find that the hyperuniformity of the DNA
droplets reflects near-equilibrium dynamics, where phase separation drives the
organization of droplets that then relax toward equilibrium via droplet
Brownian motion. We engineer a two-species system of immiscible DNA droplets
and find two distinctly hyperuniform structures in the same sample, but with
random cross-species droplet correlations, which rules out explanations that
rely on droplet-droplet hydrodynamic interactions. In addition, we perform
experiments on the electrostatic coacervation of peptides and nucleotides which
exhibit hyperuniform structures indistinguishable from DNA nanostars,
indicating the phenomenon generally applies to phase-separating systems that
experience Brownian motion. Our work on near-equilibrium droplet assembly and
structure provides a foundation to investigate droplet organizational
mechanisms in driven/biological environments. This approach also provides a
clear path to implement phase-separated droplet patterns as exotic optical or
mechanical metamaterials, or as efficient biochemical reactors.
