Hagan Group

Motility-Driven Phase Separation of Repulsive Self-Propelled Spheres in 2D

We examined a minimal model for an active colloidal fluid in the form of self-propelled Brownian hard spheres that interact purely through excluded volume. Using simulations and analytic modeling, we quantified the phase diagram and separation kinetics.

We found that this nonequilibrium active system undergoes an analog of an equilibrium continuous phase transition, with a binodal curve beneath which the system separates into dense and dilute phases, whose concentrations depend only on activity (Fig. 1).  The dense phase is a unique material that we call an active solid, which exhibits the structural signatures of a crystalline solid near the crystal-hexatic transition point, and anomalous dynamics including superdiffusive motion on intermediate timescales (Fig. 2). This work was published in [1].

Motivated by recent experiments [2], we generalized the study to consider a system of self-propelled colloids that experience short-range attractive interactions and are confined to a surface [3]. In this case, we found that the phase behavior for such a system is reentrant as a function of activity: phase-separated states exist in both the low- and high-activity regimes, with a homogeneous active fluid in between.  We developed a kinetic model for the system's steady-state dynamics, whose solution captures the main features of the phase behavior and reveals the physical mechanisms that give rise to reentrance. The simulated phase diagram can be directly tested against experiments on self-propelled particle with attractions.

spheresFig. 1: Computer simulations predict that self-propelled spheres in 2D undergo phase separation. On the left, each line shows the behavior of a system at a different area fraction (f) of self-propelled spheres, as a function of the Peclet number (Pe) which measures the propulsion velocity normalized by diffusive noise. The dashed black line indicates the binodal, above which the system separates into a dense phase and a dilute phase.

propelledFig. 2:  Computer simulations of self-propelled spheres in 2d predict a new form of matter: an active liquid crystal. (a) The structure factor for an example active crystal showing crystallinity. (b) Instantaneous speed of particles within the system, illustrating the inhomogeneous motion within the crystal. (c) Particles in the active crystal undergo diffusion. The image shows the analog of a FRAP experiment. The particles which start in the center are labeled blue; the image to the right shows a snapshot after evolution of the dynamics.


  1. Redner, G.S., M.F. Hagan, and A. Baskaran, Structure and Dynamics of a Phase-Separating Active Colloidal Fluid. Phys. Rev. Lett., 2013. 110(5):  055701.
  2. Palacci, J., S. Sacanna, A.P. Steinberg, D.J. Pine, and P.M. Chaikin, Living Crystals of Light-Activated Colloidal Surfers. Science, 2013. 339(6122):  936-940.
  3. Redner, G.S., A. Baskaran, and M.F. Hagan, Reentrant Phase Behavior in Active Colloids with Attraction. Phys. Rev. E, 2013. 88:  012305.