Phoretic Interactions Generically Induce Dynamic Clusters and Wave Patterns in Active Colloids

We introduce a representative minimal model for phoretically interacting active colloids. Combining kinetic theory, linear stability analyses, and a general relation between self-propulsion and phoretic interactions in autodiffusiophoretic and autothermophoretic Janus colloids collapses the parameter space from six to two dimensionless parameters: area fraction and Péclet number. This collapse arises when the lifetime of the self-generated phoretic fields is not too short, and leads to a universal phase diagram showing that phoretic interactions generically induce pattern formation in typical Janus colloids, even at very low density. The resulting patterns include waves and dynamic aggregates closely resembling the living clusters found in experiments on dilute suspension of Janus colloids.

Figure 1

Generic patterns in phoretic Janus colloids: snapshots from particle-based simulations (movies and parameters in the Supplemental Material [8]). (a–c, movie 1) Attractive phoretic interactions induce clusters at early times (s) which merge (b), accompanied by a colocated phoretic field (c). (d–f, movie 5) Dynamic clusters induced by repulsive phoretic interactions (d) surrounded by shells of large phoretic fields (e) which do not coarsen beyond a characteristic size (f). (g–l, movies 2–4) Colloidal waves (g) pursued by self-produced phoretic waves caging the colloids in dense clusters (h); these clusters act as enhanced phoretic producers leading to phoretic clusters (i) which drive colloids away, and induce escape waves (j). At late times, these wave patterns may settle into regular moving bands of colloids closely followed by phoretic waves (k,l).

Figure 2

First set of four panels (from left): classification of autophoretic Janus colloids by their response to phoretic gradients (attractive and repulsive) and their swimming direction (cap ahead and behind). Second set of four panels: sketch of the response of Janus colloids to the fields produced by other colloids with indications of which sign coefficients are required for the Keller-Segel (KS), Janus (Janus), and delay-induced (delay) instability. Right figure: universal phase diagram for quasi-2D repulsively interacting autophoretic colloids depending only on Péclet number (Pe) and area fraction ($f$). Lines show the onset of the delay-induced (blue) and Janus instability (red); the latter illustrates the case where colloids produce and consume and requires twice as large Pe numbers for pure producers; see [8].