Effective Cahn-Hilliard Equation for the Phase Separation of Active Brownian Particles

Abstract

The kinetic separation of repulsive active Brownian particles into a dense and a dilute phase is analyzed using a systematic coarse-graining strategy. We derive an effective Cahn-Hilliard equation on large length and time scales, which implies that the separation process can be mapped onto that of passive particles. A lower density threshold for clustering is found, and using our approach we demonstrate that clustering first proceeds via a hysteretic nucleation scenario and above a higher threshold changes into a spinodal-like instability. Our results are in agreement with particle-resolved computer simulations and can be verified in experiments of artificial or biological microswimmers.PACS numbers: 82.70. Dd,64.60.Cn The collective behavior of living "active" matter has recently attracted considerable interest from the statistical physics community (for reviews, see Refs. 1, 2). Even if the mutual interactions of the individual units are following simple rules, complex spatiotemporal patterns can emerge. Examples in nature occur on a wide range of scales from flocks of birds [3] to bacterial turbulence [4]. A basic physical model is obtained by describing the individual entities as particles with internal degrees of freedom (in the simplest case just an orientation) that consume energy and are thus driven out of thermal equilibrium. Consequently, shaken granular particles [5] and phoretically propelled colloidal particles [6-9] have been investigated in detail. Moreover, the observed collective behavior might find applications in, e.g., the sorting [10] and transport of cargo [11].Here we are interested in the phase behavior of repulsive particles below the freezing density. While in equilibrium only one fluid phase exists, sufficiently dense suspensions of repulsive self-propelled disks undergo an "active phase separation", i.e., particles aggregate into a dense, transiently ordered cluster surrounded by a dilute gas phase. This has been observed first [12,13] in computer simulations of a minimal model [14][15][16]. Clustering has also been reported in experiments using colloidal suspensions of active Brownian particles, in which the particles are phoretically propelled along their orientations due to the catalytic decomposition of hydrogen peroxide on a platinum hemisphere [7], or due to lightactivated hematite [8]. In these experiments, phoretic attractive forces play an important role. A closer realization of ideally repulsive particles is possible through the reversible local demixing of a near-critical water-lutidine mixture [17]. Colloidal particles propelled due to the ensuing local density gradients show indeed the predicted phase separation [9]. While in passive suspensions phaseseparation occurs only for sufficiently strong attractive forces, the microscopic mechanism for repulsive active particles is due to self-trapping: colliding particles block each other due to the persistence of their orientation [9]. In sufficiently dense suspensions, the "pressure" of the free, fast particles leads to the...

Shape control and compartmentalization in active colloidal cells

“…In the continuum model, the spinner dynamics is described by coupling the Cahn-Hilliard phase field equation to a Navier-Stokes equation with an active term representing the rotational driving torque. Previously, a continuum model was used to describe separation of translationally driven particles into high-and low-density phases, much like vapor-liquid or vapor-solid coexistence in single-component equilibrium systems (20,61,62). Here, instead, we model separation into clockwiseand counterclockwise-driven domains, analogous to equilibrium phase separation of a binary mixture of immiscible fluids as reported in ref.…”

Focusing by blocking: Repeatedly generating central density peaks in self-propelled particle systems by exploiting diffusive processes

“…The nature of steric [41,42] as well as of possible hydrodynamic interactions [43,44] depends on details of the particular system under consideration. To remain at this stage as general as possible, we only consider the most central feature of hard steric pair interactions: a diverging interaction energy when two particles are found at the same position.…”

Lane formation in a driven attractive fluid

“…To this end we define an effective (equilibrium) model system in which the origi-nal pair interactions are corrected by the driving force. Similar attempts to map an intrinsically non-equilibrum system onto an effective equilibrium one have been done in the context of active particles, involving effective pair potentials [9,12,[31][32][33], an effective Cahn-Hillard equation [34] or non-equilibrium equations of states [35].…”