Fig. 1: Activity-induced phase transition.
From: Shaping active matter from crystalline solids to active turbulence
a N circular phoretic disks of radius a freely swimming in a periodic square domain of size \(\tilde{L}\). The notation ̃ represents dimensional variables throughout. The side panel illustrates the numerical discretization (Supplementary information Sec. II). b sketch of a single phoretic disk attaining a swimming velocity, \(\tilde{{{{{{{{\bf{U}}}}}}}}}\), spontaneously via instability. Green dots indicate a chemical solute emitted by the disk, while \({\tilde{{{{{{{{\bf{u}}}}}}}}}}_{{{{{{{{\rm{slip}}}}}}}}}\) denotes its surface slip velocity induced by the chemical gradient. c A swimming disk of low (left) or high (right) activity—implied by the Péclet number Pe—follows a straight or chaotic trajectory, respectively. The colormap shows the scaled solute concentration \(c/{c}_{\max }\) and the insets depict the trajectories. The streamlines surrounding the swimmer at Pe = 2.5 demonstrates its pusher-like dipolar signature, as previously identified numerically31 and experimentally88. d–f Disks with an area fraction ϕ = 0.12 self-organize into hexagonal solid (Pe = 2), liquid (Pe = 2.5), and gas-like (Pe = 3) phases depicted in a quarter of the domain. Disks are colored by their swimming speed ∣U∣ and the arrow indicates the instantaneous direction of U. The second row displays the corresponding pair correlation function \(g({{{{{{{\mathcal{R}}}}}}}})\). Here, \({{{{{{{\mathcal{R}}}}}}}}\) denotes the inter-disk distance and \(\ell={\left(2\pi {\phi }^{-1}/\sqrt{3}\right)}^{1/2}\) represents the lattice constant. g, h Root-mean-square (RMS) disk velocity Urms versus time and the scaled mean square displacement (MSD) versus the time lag τ, respectively, for varying Pe. See Source data.
