Instability and spinodal decomposition of chemically active suspensions

Chemically active particles can self-propel by diffusiophoresis with velocity $\mathbf{U}=-M\nabla c$ by changing the local solute concentration $c$ via a surface catalytic reaction. Here, $M$ is the particle dffusiophoretic mobility. The particle and fluid motion is such that the convection of solute can be ignored and the concentration field $c$ is governed by Laplace's equation. We explore the collective dynamics of active particles by both continuum theory and particle-tracking simulation. In simulation the solute concentration field is accurately resolved simultaneously with the particles' motion by a multipole scattering method allowing the simulation of thousands of active particles. Active suspensions exhibit a Brinkman-like screening of long-range interactions which predicts an instability in the collective dynamics that scales with the volume fraction of active particles to the $1/2$ power. For weak phoretic motion (small $M$), the instability theory is verified by the simulations. For strong phoretic motion (large $M$), the active particles show a spinodal decomposition. Transient fractal structures are identified in 3D, while individual clusters are observed in a particle monolayer.