Motility-induced phase separation is reentrant

Active systems of self-propelled units, from microorganisms to artificial colloids, operate far from equilibrium and display collective behaviors without equilibrium analogs. A minimal model is active Brownian particles (ABPs) with purely repulsive interactions, which can undergo motility-induced phase separation (MIPS) into dense and dilute phases due to activity-induced effective attractions. Prior work has shown the phase diagram of ABPs resembles that of equilibrium attractive fluids, with activity playing a role similar to inverse temperature. This raises the question of whether activity can induce unique phase behaviors beyond mimicking equilibrium temperature. The study addresses whether activity alone can cause reentrant MIPS—phase separation that appears and then disappears upon further increasing activity—thus revealing an intrinsically nonequilibrium mechanism absent in equilibrium systems.

Foundational studies established MIPS in active systems such as run-and-tumble and ABP models and developed kinetic and thermodynamic-like descriptions (e.g., Cates & Tailleur; Redner et al.; Speck et al.; Takatori & Brady). For repulsive ABPs, condensation at interfaces balances rotational-diffusion-driven evaporation, producing binodal predictions matching simulations at small to moderate activity. Prior comparisons emphasized the analogy to equilibrium phase diagrams with attraction, effectively mapping activity to inverse temperature. Additional works explored nucleation pathways, spinodal decomposition, swim pressure, and hydrodynamics. However, equilibrium analogs do not exhibit reentrant phase separation for simple attractive interactions, and earlier active-matter reentrance involved explicit attractive potentials, not purely repulsive ABPs. This study builds on kinetic theories and dense-phase transport analyses to incorporate an activity-dependent diffusional contribution to evaporation, enabling reentrance.

The work combines a kinetic theory of MIPS with numerical simulations of 2D repulsive ABPs.

• Kinetic theory: The steady state of coexistence is modeled with a dense (close-packed) phase and a dilute homogeneous isotropic phase. Particles exchange between phases via condensation (rate k_in) and evaporation (rate k_out). Condensation current is j_in = ρ_a V0/π, where ρ_a is dense-phase number density and V0 the self-propulsion speed. Traditional models take k_out ∝ D_r (rotational diffusion), adequate at small to moderate activity. Here, the authors add a diffusional current proportional to the density gradient with an effective dense-phase diffusion coefficient D = D_t + λ D_s/( (ρ_a/ρ_d)^α ), where D_t is thermal diffusivity, D_s is swim diffusivity, λ ∈ (0,1) encapsulates interaction-mediated coupling, and α parameterizes density dependence. The total outward current from the dense phase is J_out = D ∇ρ + k_out, yielding in 1D interfacial balance a steady-state condition j_in = j_out. This leads to two transition points for activity (Eq. (2) in the text), implying that increasing V0 first induces MIPS then melts it at larger V0 (reentrance). The dilute-phase density at steady state follows ρ_d = (p D + k D_r)/(D + V0 σ/π) (using κ-like fitting parameters), which increases toward ρ_a at very large activity, consistent with homogenization.
• Predictions from theory: (i) Reentrant MIPS due to competition between activity-induced effective attraction (via reduced speed at high density) and activity-induced nonequilibrium vaporization (enhanced dense-phase diffusivity). (ii) Dependence on interaction strength ε in the repulsive Weeks-Chandler-Andersen (WCA) potential U(r)=4ε[(σ/r)^12 − (σ/r)^6 + 1/4] for r<2^{1/6}σ: larger ε reduces dense-phase effective diffusivity (smaller parameter A or λ), increasing the upper transition Pe and broadening the MIPS window; softer repulsion enhances reentrance.
• Numerical simulations: 2D periodic box with N spherical ABPs (diameter σ, friction γ) interacting via purely repulsive WCA potential (as above). Overdamped Langevin dynamics: ṙ_i = γ^{-1}F_i − ∇_i V + U(r_i) + ξ_i; orientation diffuses with rotational noise η_i. Thermal translational diffusion D_t = k_B T/γ; rotational diffusion D_r = 3 D_t/σ^2. Control parameters: Péclet number Pe = V0 σ/D_t (≈ V0 σ^2/(γ D_t)) characterizing activity; area fraction φ = N π σ^2/(4 L_x L_y). Units: σ for length, 20 k_B T for energy, time unit γ σ^2/(20 k_B T); γ=1. Time step Δt between 1e−5 and 1e−6 depending on Pe; total simulation time ≈ 500 (in chosen units). Protocols: random initial conditions inside spinodal; near binodal, use forward/backward continuation in Pe to probe hysteresis. Finite-size effects checked with larger systems (Supplementary). Phase behavior extracted via binodal and spinodal from cluster statistics; fraction of dense-phase particles f_1 = (ρ_a − ρ_d)/ρ_a computed using theory with fitted parameters (e.g., λ = 2.8×10^−4, κ = 0.9 in Fig. 1).

• Reentrant MIPS: Theory predicts two activity thresholds (Eq. (2)), so that increasing activity first induces phase separation (MIPS) and then suppresses it at higher activity due to activity-enhanced dense-phase diffusivity (nonequilibrium vaporization). Simulations of purely repulsive ABPs confirm this reentrance.
• Phase diagram agreement: In the Pe–φ plane (ε=1), the simulated binodal (red dotted line) matches the theoretically predicted boundary (background color map of f_1 with λ=2.8×10^−4, κ=0.9). Both binodal and spinodal shift to lower φ with increasing Pe, then reenter to higher φ after an inflection, yielding a widened metastable region at large Pe. For φ≈0.35, the system transitions single phase → coexistence → single phase as Pe increases.
• Nucleation near upper transition: Near the upper spinodal/reentrant transition (e.g., φ=0.35, Pe≈900), the largest cluster fraction N_max/N exhibits a waiting time t_nuc before rapid growth, consistent with first-order nucleation. The ensemble-averaged waiting time ⟨t_nuc⟩ increases exponentially with Pe above the upper spinodal, indicating an increasing nucleation barrier. A pronounced hysteresis loop around the upper transition further supports discontinuous behavior.
• Interaction-strength dependence: In the Pe–ε plane at fixed φ=0.3, the upper binodal increases markedly with ε, whereas the lower binodal is nearly unchanged. Steady-state configurations at Pe=2500 show that increasing ε from 1.0 to 4.0 drives more robust MIPS (larger dense clusters). Thus, harder (larger ε) repulsion weakens reentrance (broadens MIPS), while softer repulsion strengthens reentrance.
• Dilute-phase density trend: From the steady-state expression ρ_d = (p D + k D_r)/(D + V0 σ/π), ρ_d decreases with V0 at low to moderate activity but increases toward ρ_a at very large activity, implying homogenization and corroborating reentrance.
• Conceptual advance: Identifies an activity-induced nonequilibrium vaporization mechanism—distinct from equilibrium analogs—that competes with activity-induced effective attraction, fundamentally altering phase behavior.

The findings demonstrate that activity in repulsive ABPs plays a dual role: it promotes clustering by slowing particles in dense regions (effective attraction) while simultaneously enhancing escape via increased dense-phase diffusivity (nonequilibrium vaporization). The kinetic balance between condensation and this composite evaporation current yields two activity thresholds and a reentrant phase diagram, in stark contrast to equilibrium systems with simple attractive interactions. The excellent agreement between kinetic theory and simulations across Pe–φ and Pe–ε planes supports the physical picture and highlights that dense-phase transport properties, not just interface rotation, control MIPS at high activity. The dependence on interaction strength shows how microscopic repulsion modulates dense-phase diffusivity and thus the window of MIPS, offering a knob to tune or suppress reentrance. These results refine our understanding of active phase separation by emphasizing intrinsically nonequilibrium transport within dense phases and may inform design strategies for active materials where phase behavior is controlled by activity and interaction softness.

The study uncovers reentrant MIPS in purely repulsive ABPs, arising from competition between activity-induced effective attraction and an activity-driven nonequilibrium vaporization process that accelerates melting at high activity. A kinetic theory incorporating dense-phase diffusional currents predicts two activity thresholds, reentrant phase boundaries, and their dependence on interaction strength. Large-scale simulations confirm these predictions, including binodal/spinodal structures, nucleation and hysteresis near the upper transition, and the trend with ε. This highlights a uniquely nonequilibrium mechanism with no analog in passive equilibrium fluids. Potential future work includes: quantifying dense-phase diffusivity and its microscopic origins across interaction models; testing reentrant MIPS in experimental active colloids or bacterial systems; extending to 3D, hydrodynamic interactions, or anisotropic particles; and developing continuum theories that self-consistently couple activity, stress, and dense-phase transport.

• The kinetic theory uses phenomenological parameters (e.g., λ, κ, A/p,k) and assumes a close-packed dense phase and homogeneous dilute phase; quantitative predictions rely on fitting and may not capture all microscopic details.
• The separation between rotational-diffusion-driven evaporation and diffusional escape is acknowledged to be ambiguous, potentially leading to double counting; λ is introduced to mitigate this empirically.
• Simulations are in two dimensions with WCA repulsion and overdamped dynamics without explicit hydrodynamics; generalization to other interactions, dimensionalities, or hydrodynamic couplings requires further validation.
• Finite-size effects are considered but may still influence nucleation statistics and binodal estimates; larger-scale studies could refine boundaries.