Introduction

Active systems consisting of self-propelled units have been widely discovered in nature on many scales, ranging from mesoscopic biological or manmade swimmers such as E. coli and Janus spheres to macroscopic objects like fish, birds and horses1,2. Due to the ability to take in and dissipate energy to drive themselves far from equilibrium3, active systems provide ideal model systems to investigate the effect of nonequilibrium on collective behaviors beyond their equilibrium counterparts4,5,6,7,8,9,10,11,12,13,14,15,16,17,18. As one of the simplest active systems, active Brownian particles (ABPs) with pure repulsion without any attraction can spontaneously undergo phase separation between dense and dilute fluid phases19,20. Such phase separation resulting solely from the intrinsically nonequilibrium property, i.e., activity, is so-called motility-induced phase separation (MIPS)16,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34.

Generally, for ABPs with pure repulsion, they tend to accumulate where they move more slowly and will slow down at high density for steric reasons, which then creates effective attractions leading to MIPS20. Quite interestingly, it has been found that the phase diagram of ABPs20,21,22,23 is nearly the same as that of the simplest equilibrium system with attractive inter-particle interactions35,36,37, except that the role of (inverse) temperature is replaced by activity. Is that all about the effect of activity on the phase separation of ABPs? Will the nonequilibrium nature of activity bring unique phase behaviors to the simplest ABPs beyond providing an alternative origin for phase separation?

In this work, we report that activity can lead to a unique phase behavior of a counterintuitive MIPS reentrance, which lacks an analog in equilibrium systems. Our theoretical analysis based on a kinetic theory of MIPS in the simplest ABP system with purely repulsive interactions shows that, this phase behavior stems from a new type of nonequilibrium vaporization induced by activity. While activity-induced effective attraction leads to MIPS, activity-induced nonequilibrium vaporization hinders the formation of dense phase and thus results in the MIPS reentrance. Besides, we find that the “softer" the repulsive interaction potential is, the “stronger" the reentrance becomes. In simulations, the reentrant MIPS is confirmed and fits very well with our theory. Specifically, the simulated binodal agrees well with the phase boundary derived by our theoretical analysis, and both simulations and theory give the same relationship between reentrant MIPS and the strength of interactions.

Results and discussion

Theoretical analysis

We start from a minimal active fluid theory characterizing MIPS from a kinetics approach21,22,38. The theory describes the steady state of phase separation with a dense clustering phase which is set as close packed and a dilute phase as homogeneous and isotropic. Particles transport between two phases through condensation and evaporation events with rates kin and kout, respectively. Absorption rate kin is assumed to be proportion to the dilute phase number density ρg and the characteristic self-propulsion velocity v0. For the vaporization, the theory sets that active particles immediately escape from the dense phase when the direction of the particle on the interface moves towards to dilute phase. Therefore the escape rate kout is proportion to the rotational diffusion constants Dr. Then the steady state condition is achieved through equating kin and kout, leading to the dilute phase density \({\rho }_{g}=\frac{\pi {\kappa }^{{\prime} }{D}_{r}}{\sigma {v}_{0}}\) where \({\kappa }^{{\prime} }\) is a fitting parameter and σ is the particle diameter. For two dimensional ABP system with purely repulsive inter-particle potential21,22 and Lennard–Jones potential38, solutions of ρg show very good agreement with simulated MIPS. However, it should be noted that the particle current from cluster to dilute phase comprises not only vaporization events due to rotational-diffusion-dependent active motion, but also diffusion process along the particle density gradient. Good agreement between solutions of ρg and simulated MIPS indicates the former one makes a major contribution to kout for activity ranging around the transition point of MIPS (i.e., small and moderate activity). Yet for large activity, the contribution of diffusion process becomes progressively more important as will be shown by the following analysis.

In general, the diffusion current is proportional to the density gradient ∇ ρ with coefficient D. In the gas phase, the translational diffusion coefficient is close to the effective one for a free active particle \({D}_{0}={D}_{t}+\frac{{v}_{0}^{2}}{2{D}_{r}}\), where Dt is purely thermal diffusion constant20 and the second term of rhs is so called “swim diffusivity"39. However, in the dense phase, this diffusion coefficient D is not simply Dt nor D0. Inspired by refs. 40,41, wherein an “instantaneous diffusion coefficient" for selected particle-i was introduced to investigate the time evolution of dense ABP system, defined as \({D}_{i}={D}_{t}+\frac{{v}_{0}^{2}/(2{D}_{r})}{1+\frac{1}{2{D}_{r}\gamma }{\sum }_{j\ne i}{\nabla }_{i}^{2}U({r}_{ij})}\), where U(rij) is the inter-particle potential and γ is the friction coefficient. Although this Di is not a long-time diffusivity that one we desired, it brings a good phenomenological method to estimate D. Based on the formula of Di, we write the effective diffusion in dense phase as \(D={D}_{t}+\frac{\lambda {v}_{0}^{2}}{{D}_{r}}\) in the sense of average, with a dimensionless parameter \(\lambda \in (0,\frac{1}{2})\) representing the complex interaction. We do notice that the divide of evaporation event and diffusional process, is a little ambiguous and there might be double counting in this context. At least for a single event that a particle leaves the dense phase, it is difficult to determine which type of diffusion plays a major role. Nevertheless, we use a fitting parameter κ to compromise this defect, then the total current leaving dense phase can be written as

$${j}_{{{{{{{{\rm{out}}}}}}}}} =D\nabla \rho +\kappa \frac{{D}_{r}}{\sigma }\\ =\left({D}_{t}+\frac{\lambda {v}_{0}^{2}}{{D}_{r}}\right)\frac{{\rho }_{d}-{\rho }_{g}}{\sigma }+\kappa \frac{{D}_{r}}{\sigma },$$
(1)

where ρd is the number density of the dense phase. Analogous to the procedure in ref. 21, by integrating the angles of ABPs over the direction of self-propulsion toward the interface, the condensation current is written as jin = ρgv0/π.

Based on the steady-state assumption, i.e., jin = jout, two transition points can be immediately derived as

$${v}_{c}^{\pm }=\frac{{D}_{r}\sigma }{2\lambda ({\rho }_{d}-{\rho }_{g})}\left[\frac{{\rho }_{g}}{\pi }\pm \sqrt{\frac{{\rho }_{g}^{2}}{{\pi }^{2}}-4\frac{\lambda ({\rho }_{d}-{\rho }_{g})}{{D}_{r}{\sigma }^{2}}\left[{D}_{t}({\rho }_{d}-{\rho }_{g})+\kappa {D}_{r}\right]}\right].$$
(2)

We recall that, in the previous understanding of MIPS taking only rotational-diffusion-dependent vaporization into account, there is only one transition point vc, since jout is independent on v0 and jin increases linearly with v021. Consequently, MIPS will be observed as v0 > vc, and activity is considered to mainly provide a nonequilibrium origin for phase separation. Quite interestingly, Eq. (2) points out that, increasing activity not only leads to MIPS when v0 passes \({v}_{c}^{-}\) but also destroys it when \({v}_{0} \, > \, {v}_{c}^{+}\). Such reentrant MIPS of the simplest ABPs with purely repulsive inter-particle interactions indicates that, beyond the activity-induced effective attraction when only rotational-diffusion-dependent vaporization is considered, an extra effective diffusivity due to active motion further contributes a contrary effect on phase separation by accelerating melting. Such unique phase behavior induced by the nonequilibrium nature of activity has never been observed in the equilibrium analogs with attractive inter-particle interactions.

There are several further predictions can be made about the interesting phase behavior. Firstly, the steady state assumption jin = jout gives a quantitative measurement of the dilute phase density, \({\rho }_{g}=\left({\rho }_{d}D+\kappa {D}_{r}\right)/\left(D+{v}_{0}\sigma /\pi \right)\). Unlike the previous theory where dilute phase density decreases with v0 non-monotonically, ρg eventually increases to ρd for very large activity. This result strongly implies a homogeneous state when activity is sufficiently large, which confirms the reentrance of MIPS from another point of view.

Secondly, the parameter λ should be dependent on the repulsive inter-particle potential \(U(r)=\epsilon \hat{U}(r)\), where ϵ is the strength of the potential and \(\hat{U}(r)\) is a dimensionless inter-particle potentials. Qualitatively, larger ϵ brings stronger repulsive force which hinders particle motion in the dense phase, and consequently leads to smaller λ and smaller diffusivity D. According to Eq. (2), \({v}_{c}^{+}\) prominently increases with interacting strength ϵ, while on the contrary \({v}_{c}^{-}\) slowly decreases. As a result, MIPS would also be reentrant for finite interacting strength ϵ, while increasing ϵ would greatly broaden the parameter region for emergence of MIPS. It is thus another testable prediction of our theory to investigate the dependence of MIPS behavior on the interacting strength.

Numerical simulation

In order to verify the unique phase behavior predicted by the aforementioned theoretical analysis, we perform simulations of a two-dimensional system with size Lx × Ly and periodic boundary conditions consisting of N spherical ABPs with diameter σ and friction coefficient γ. The pairwise inter-particle potential is set as Weeks–Chandler–Andersen (WCA) potential \(U({r}_{ij})=4\epsilon [{(\sigma /{r}_{ij})}^{12}-{(\sigma /{r}_{ij})}^{6}+1/4]\) for rij < 21/6σ and U(rij) = 0 otherwise. We set the translational diffusion coefficient Dt = kBT/γ so that the system satisfies the fluctuation-dissipation relation with kB the Boltzmann constant and T the temperature. We presume that the rotational diffusion coefficient Dr is coupled with the translational diffusivity as Dr = 3Dt/σ2. In this work, we use two dimensionless variables as control parameters, i.e. the Péclet number Pe = v0σ/Dt21 to characterize the activity of ABPs and the area fraction ϕ = πρ0σ2/4 to describe the density of ABPs with ρ0 = N/(Lx × Ly) the averaged number density of the system. In simulations we set σ as the unit of length, 20kBT as the unit of energy, and γσ2/(20kBT) as unit of time. (See more details of the model in the Method section).

The phase diagram in the Pe − ϕ plane with interaction strength ϵ = 1 is presented in Fig. 1a, wherein the color bar denotes the theoretical prediction of particle number fraction in dense phase \({f}_{l}=\frac{{\rho }_{d}}{{\rho }_{0}}\frac{{\rho }_{0}-{\rho }_{g}}{{\rho }_{d}-{\rho }_{g}}\) (since the number density of dense phase ρd is not sensitive to the Péclet number and total density ρ0, we assume that ρd is constant and therefore, fl only depends on ρg and ρ0). It can be found that the simulated binodal (red dotted line) confirms with the phase boundary of the colored background predicted by our theoretical analysis. Likewise, theoretically predicted reentrance of MIPS can be observed as activity increases. For appropriate ϕ such as ϕ = 0.35, the active system changes from a single phase to a coexisting phase and reenters to a single phase as Pe increases (typical snapshots can be found in Supplementary Fig. S1 in Supplementary Discussion I). Besides of the binodal, the spinodal (green dotted line) can also be obtained by simulations too. It can be found that both the binodal and spinodal curves shift to lower ϕ first, but reenter to higher ϕ again after an inflexion, forming a brand new metastable state located in a much wider region at large Pe.

Fig. 1: The reentrant MIPS.
figure 1

a Phase diagram in the Pe − ϕ plane. The colored background is obtained by the kinetics model with λ = 2.8 × 10−4 and κ = 0.9. fl is the fraction of dense phase. The red and green lines respectively represent the binodal and spinodal curves from simulations. b Dependence of the ensemble-averaged waiting time 〈tw〉 for the nucleation on Pe for active systems at ϕ = 0.35 inside the gray dashed box in a. Curves are the exponential fitting for the simulation data. The inset is the dependence of \({N}_{\max }/N\) with \({N}_{\max }\) the particle number of the largest cluster on t for ten independent ensembles at ϕ = 0.35 and Pe = 900. c Hysteresis at ϕ = 0.35 including the gray dashed box in a. Simulation data are reached from two initial conditions: the final configurations of systems at slightly smaller Pe (red line and symbols) and the ones at slightly larger Pe (blue line and symbols).

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To investigate the nature of MIPS near the MIPS reentrance region, the growth process of the largest cluster (Fig. 1b) and a hysteresis loop (Fig. 1c) are focused. The inset of Fig. 1b plots time series of \({N}_{\max }/N\) with \({N}_{\max }\) the particle number of the largest cluster for ten independent simulations with ϕ = 0.35 and Pe = 900 (time series for other parameters are shown in Supplementary Fig. S2 in Supplementary Discussion II). It can be observed that \({N}_{\max }/N\) increases from about 0 quickly after a waiting time tw, indicating a first-order phase transition via nucleation21. Noticing that the waiting time can be a good parameter to measure the nucleation barrier, the obtained ensemble-averaged waiting time 〈tw〉 as a function of Pe is then presented in Fig. 1b. As Pe increases across the upper spinodal, 〈tw〉 increases exponentially, demonstrating that systems with larger Pe above the upper spinodal must go over a higher nucleation barrier to attain MIPS. Besides, a very large hysteresis loop with ϕ = 0.35 can be found around the upper MIPS transition point as shown in Fig. 1c, demonstrating again the nucleation behavior of a discontinuous transition from a homogeneous initial state to MIPS24. In short, the simulation results are consistent with our theoretical predictions, verifying that activity can bring unique phase behavior to the simplest ABP system.

Figure 2 a verifies the theoretical prediction about the effect of interaction strength ϵ on MIPS with fixed ϕ = 0.3, where blue and red symbols indicate the single phase and the coexisting phase of MIPS, respectively. Phase boundaries between these two phases are the binodal (red lines), and the colored background is fl obtained by our theoretical analysis. Clearly, the upper binodal increases remarkably as ϵ increases, while the lower binodal stays nearly unchanged. Such observation agrees with the theoretical prediction very well. To take a close look at the influence of ϵ, steady-state configurations of systems at Pe = 2500 and ϵ = 1.0, 1.5, 2.0 and 4.0 are shown in Fig. 2b–e, respectively. As ϵ increases, the dense phase emerges with a relatively small cluster, and then grows to be a large cluster, indicating more and more pronounced MIPS. In other words, “harder" repulsive interactions between ABPs will lead to “weaker" reentrant MIPS while the “softer" ones will result in “stronger" reentrant MIPS, which further demonstrates the unique phase behavior induced by the nonequilibrium nature of activity.

Fig. 2: Influence of the interaction strength.
figure 2

a Phase diagram in the Pe − ϵ plane at ϕ = 0.3. The colored background is obtained by the kinetics model. The blue symbols represent the single phase, the red ones denote MIPS and the red-line boundaries is the binodal curves, which are all obtained by simulations. The steady-state configurations at Pe = 2500 and b ϵ = 1.0, c 1.5, d 2.0 and e 4.0 are respectively marked as green stars in a.

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Conclusion

In summary, reentrant MIPS induced solely by activity has been revealed. We showed both theoretically and numerically that such reentrant MIPS results from the competition between activity-induced effective attraction of ABPs preferring particle accumulation and activity-induced nonequilibrium vaporization hindering formation of large clusters. Our findings highlight the unique role played by the nonequilibrium nature of activity on phase behaviors of active systems, which may inspire deep insights into the essential difference between equilibrium and nonequilibrium systems.

Method

We consider a two-dimensional system with size Lx × Ly and periodic boundary conditions consisting of N spherical ABPs with diameter σ and friction coefficient γ. The motion of the i-th ABP located at ri obeys the following overdamped Langevin equations:

$${\dot{{{{{{{{\bf{r}}}}}}}}}}_{i}={v}_{0}{{{{{{{{\bf{n}}}}}}}}}_{i}-{\gamma }^{-1}\mathop{\sum }\limits_{j=1,j\ne i}^{N}{\nabla }_{{{{{{{{{\bf{r}}}}}}}}}_{i}}U({r}_{ij})+{{{{{{{{\boldsymbol{\xi }}}}}}}}}_{i},$$
(3)
$${\dot{{{{{{{{\bf{n}}}}}}}}}}_{i}={{{{{{{{\bf{n}}}}}}}}}_{i}\times {{{{{{{{\boldsymbol{\eta }}}}}}}}}_{i}.$$
(4)

Herein, Eq. (3) represents the translational motion of the i-th ABP where v0 and ni is respectively the amplitude and direction of active velocity. The interaction between a pair of ABPs is described by the purely repulsive Weeks–Chandler–Andersen (WCA) potential: \(U({r}_{ij})=4\epsilon [{(\sigma /{r}_{ij})}^{12}-{(\sigma /{r}_{ij})}^{6}+1/4]\) for rij = ∣ri − rj∣ < 21/6σ, and U(rij) = 0 otherwise, with ϵ the interaction strength. The last term ξi denotes the Gaussian white noise vector with time correlations \(\langle {{{{{{{{\boldsymbol{\xi }}}}}}}}}_{i}(t){{{{{{{{\boldsymbol{\xi }}}}}}}}}_{j}({t}^{{\prime} })\rangle =2{D}_{t}{{{{{{{\bf{I}}}}}}}}{\delta }_{ij}\delta (t-{t}^{{\prime} })\), where Dt is the translation diffusion coefficient and I is the unit matrix. We set Dt = kBT/γ so that the system satisfies the fluctuation-dissipation relation when propulsion forces are absent. Herein, kB denotes the Boltzmann constant and T represents the temperature. Eq. (4) is the rotational equation of the i-th ABP. For a two-dimensional system, ni is confined into a two-dimensional space and only z-component (perpendicular to this plane) of the white noise ηi is concerned, which has time correlation \(\langle {\eta }_{i}^{z}(t){\eta }_{j}^{z}({t}^{{\prime} })\rangle =2{D}_{r}{\delta }_{ij}\delta (t-{t}^{{\prime} })\) where Dr is the rotational diffusion coefficient which couples with the translational diffusivity Dr = 3Dt/σ2.

In simulations, we use σ and 20kBT respectively as basic units for length and energy, and set γ = 1 so that the basic unit for time is the γσ2/(20kBT). We fix Lx = 800, Ly = 200, ϵ = 1, if not otherwise stated. In this work, we use two dimensionless parameters: the Péclet number Pe = v0σ/Dt21 to characterize the activity of ABPs and the area fraction ϕ = πρ0σ2/4 to describe the density of ABPs with ρ0 = N/(Lx × Ly) the averaged number density of the system. For consistency, all of the following results are obtained from simulations of total time ttot = 500 with the time step Δt changing from 10−5 to 10−6 for different activities. Notice that this volume fraction is only relevant on a parametric level, and does not correspond to the specific values of the hard sphere system. For active systems with Pe and ϕ located inside the spinodal curve, we ran simulations from random initial conditions. Whereas, for systems close to the binodal, as Pe increases(decreases), we used the final configurations of systems with slightly smaller(larger) Pe as the initial conditions for the ones with larger(smaller) Pe21,24. In order to avoid finite-size effects especially at the large activity region, we have also made a number of simulations for larger systems, which are shown in Supplementary Discussion III.