Spontaneous demixing of chiral active mixtures in motility-induced phase separation

The demixing and sorting strategies for chiral active mixtures are crucial to the biochemical and pharmaceutical industries. However, it remains uncertain whether chiral mixed particles can spontaneously demix without the aid of specific strategies. In this paper, we investigate the demixing behaviors of binary mixtures in a model of chiral active particles to understand the demixing mechanism of chiral active mixtures. We demonstrate that chiral mixed particles can spontaneously demix in motility-induced phase separation (MIPS). The hidden velocity alignment in MIPS allows particles of different types to accumulate in different clusters, thereby facilitating separation. There exists an optimal angular velocity or packing fraction at which this separation is optimal. Noise (translational or rotational diffusion) can promote mixture separation in certain cases, rather than always being detrimental to the process. Since the order caused by the hidden velocity alignment in this process is not global, the separation behavior is strongly dependent on the system size. Furthermore, we also discovered that the mixture separation caused by MIPS is different from that resulting from explicit velocity alignment. Our findings are crucial for understanding the demixing mechanism of chiral active mixtures and can be applied to experiments attempting to separate various active mixtures in the future.


Introduction
Active matter has been garnering considerable attention over the past two decades due to its undeniable nonequilibrium nature, as well as potential applications in microdevices and smart materials. Active matter systems consist of particles that can extract energy from their environment, converting it into mechanical work at the single particle level [1]. So far, the most attention has been paid to ensembles of linearly translating active particles; however, recently, there has been an increasing interest in the study of chiral active matter . Chiral active matter involves particles that not only self-propel, but rotate as well. This type of chirality has its origin in either the shape or propulsion mechanism of the particles, causing them to move in circles in two dimensions or helices in three dimensions. Perhaps the most notorious examples of chiral active matter are those that mimick biological circular swimmers: helically swimming sperm cells [25][26][27], bacteria which move in circles near walls and interfaces [28][29][30][31], and the malaria parasite [32].
In previous work, sorting of chiral active mixtures usually requires some special external or internal conditions. However, it is still unclear whether chiral active mixtures can be spontaneously separated without special external or internal conditions. To address this, we studied chiral active particles in a two-dimensional box with periodic boundary conditions. Our results showed that a spontaneous chiral separation of the binary mixture could occur without the need for special strategies. Specifically, we found that motility-induced phase separation (MIPS) is a necessary condition for this separation. This is due to its hidden velocity alignment, whereby different particles accumulate in distinct clusters, thus facilitating the separation. We also found that the optimum angular velocity or packing fraction allows for maximum separation. There exists a finite translational or rotational diffusion at which the mixture can be maximally separated. The localized nature of the hidden velocity alignment makes the separation process highly dependent on system size. The role of the hidden velocity alignment caused by MIPS in particle separation is different from that of explicit polar velocity alignment. In summary, this finding helps to advance our understanding of chiral mixture separation mechanisms and provides insights for related experiments.

Model and methods
We consider a binary system of chiral active particles (N/2 counterclockwise (CCW) and N/2 CW particles) in a two-dimensional box of size L × L with periodic boundary conditions. The dynamics of each particle is described by the position r i ≡ (x i , y i ) of its center and the orientation θ i (t) of the polar axis n i ≡ (cos θ i (t), sin θ i (t)). The orientation θ i is determined by the rotation diffusion, the constant torque acting on the particles (which is responsible for circular swimming). We neglect both hydrodynamic interactions among the particles and inertial terms. The dynamic of particle i thus follows the overdamped Langevin equation with v 0 self-propulsion speed and µ the mobility. The constants D 0 and D r represent the rotational and translational diffusion coefficients, respectively. ζ i (t) and ξ i (t) are unit-variance Gaussian white-noise random numbers with zero average. Ω is the angular velocity due to the constant torque acting on the particles, which also denotes the chirality difference. q i = ±1 is the sign of the rotation and determines the chirality of the particle. We define particles as the CCW particles for q i = 1 and the CW particles for q i = −1.
The term F i describes the force contribution due to steric interactions between particles. Steric interactions are modeled by the force Here we choose U(r) as the Weeks-Chandler-Anderse potential [49].
where the constants ϵ and σ determine the energy unit and the nominal particle diameter, respectively.
To quantify the spatial distribution of the binary mixture, we define the separation coefficient S based on the Voronoi tessellation [50] where N is total number of particles, n s i is the number of similar neighboring particles of particle i and n t i is the total number of neighboring particles of particle i. With this definition, the binary mixture is completely mixed when S = 0 and completely demixed when S = 1. To describe the density of particles in the box, we define the ratio of the area occupied by particles to the total available area as the packing fraction Φ = Nπσ 2 /(4L 2 ).
In order to quantify the local order caused by the hidden velocity alignment in MIPS, we use the spatial velocity correlation function Q(r) = ∑ i Q i (r)/N [51], where is the angular distance between two angles. ϕ i is the angle formed by the particle velocity with respect to the x axis. The sum runs over the neighbors within a radius of r around particle i, and M k is the number of particles in it. Q(r) quantifies partial alignment even when global polarization is absent; a value of Q(r) = 1 represents perfect velocity alignment, Q(r) = 0 represents no velocity alignment, and Q(r) = −1 represents anti-velocity alignment.
Equations (1) and (2) are integrated by using the fourth-order Runge-Kutta algorithm with an integration step time 10 −5 . The total integration is 5 × 10 4 , which is sufficient to ensure that the system can reach a nonequilibrium steady state. Particle positions are initialized using a uniform random distribution within the box, and orientations are randomly chosen over the interval [0 2π]. As the particle configuration fluctuates quasi-periodically with time, the average value of the separation coefficient is calculated over 1000τ (where τ = 2π/Ω) after the system has reached the nonequilibrium steady state. Unless otherwise stated, our simulations are performed under the parameter sets: ϵ = 10.0, σ = 1.0, µ = 1.0, and N = 4000. We have found the presented results to be robust against reasonable changes to these parameters.

Results and discussion
MIPS is a self-organized phenomenon that emerges in active matter systems, including suspensions of self-propelled particles or cells. When the speed of motile particles declines significantly with the increasing local density, a uniform suspension becomes unstable. This instability results in a phase-separated state where a dilute active gas coexists with a dense liquid exhibiting considerably reduced motility. At high densities and activities, MIPS [52,53] generally occurs in active matter without explicit local alignment interactions, whereby self-propelled particles tend to accumulate and form a dense phase. Active torques (or chiralities) can strongly influence MIPS; for example, they can significantly suppress MIPS [9,[54][55][56]. We will focus on studying the spontaneous separation of chiral active mixtures in MIPS.
In general, the separation of two types of particles must satisfy two conditions: (1) a significant difference between the two types of particles exists and results in different motion behaviors; (2)the same type of particles accumulate and different types of particles repel each other. In this model, two types of particles rotate CW and CCW respectively, thus satisfying the first condition. Since there is no explicit attractive (or polar alignment) interaction between particles, the second condition does not seem to be fulfilled. However, in MIPS, there exists the hidden velocity alignment [51,57] that satisfies the second condition. Taking advantage of this hidden velocity alignment in MIPS, the same type of particles are able to accumulate into large clusters and different types of particles repel each other, making it possible for the binary mixture to be separated.

Zero rotational and translational diffusion
In order to investigate more directly the separation mechanism of the binary mixture, we first consider the minimal model of chiral active particles, neglecting both translational and rotational diffusion (D 0 = 0 and D r = 0). Accordingly, in this section, we explore the segregation dynamics of the binary mixture in MIPS by varying the self-propulsion speed v 0 , the angular velocity Ω, and the packing fraction of the particles Φ. Figure 1 shows the typical snapshots of the binary system for different Ω at v 0 = 10.0 and Φ = 0.5. When the two types of particles are identical (Ω = 0), MIPS occurs, leading to the coexistence of an active low-density gas phase and a dense liquid phase(shown in figure 1(a)). More importantly, in the dense liquid phase, the velocities of the particles align, thus creating a local order (as shown in figure 2(a)). However, since there is no difference between the two types of particles, the system is still mixed. When the chirality difference Ω increases to a certain size (e.g. Ω = 2.0 and 3.0), different types of particles will cluster in two separate large clusters, thus the binary mixture can spontaneously demix (shown in figures 1(b) and (c)). This behavior can be attributed to the presence of velocity alignment between particles. When a CCW particle meets another CCW particle, they will rotate CCW together, with a steady stream of CCW particles joining together and eventually forming a large cluster. When a CCW particle meets a CW particle, they will separate due to their opposite directions of rotation. Moreover, similar particles in the cluster move together  (shown in figure 2(b)), which assists in the stable maintenance of the cluster. Thus, CCW particles accumulate in one large cluster, and CW particles accumulate in another, allowing the mixture to separate spontaneously. However, for large Ω (e.g. Ω = 10.0), the radius R of circular motion of a single particle is small, and the particle can only move in a small area. In this case, MIPS almost disappears, similar particles can only form small clusters (shown in figure 1(d)), thus two types of particles are mixed. The binary mixture can only separate spontaneously if the chirality difference Ω is suitable for the system. To describe the local alignment caused by MIPS, we have provided the corresponding spatial velocity correlation function in figure 2(c). When Ω is large (e.g. Ω = 10.0) and no MIPS is present, the alignment measured by Q(r) is very weak. However, in the presence of MIPS (e.g. Ω = 0, 2.0, and 3.0), a strong alignment is observed, and the correlation length of the correlation function is large.
We use the separation coefficient S defined in equation (4) to quantify the separation behavior of binary mixtures. The segregation coefficient S as a function of Ω is shown in figure 3(a) for different v 0 at Φ = 0.5. It is noted that the segregation coefficient S is a peaked function of Ω. When Ω approaches zero, the radius R = v 0 /Ω of circular motion is infinitely large, all particles moves in a straight line(two types of particles are almost identical) and so cannot be separated, resulting in S being nearly zero. When Ω approaches infinity, the radius of circular motion tends to zero, causing particles to rotate in their initial positions and the self-propulsion to become insignificant; thus, MIPS is eliminated and S is small. In essence, there is an optimal radius R op of circular motion at which the mixture can be separated to the greatest extent. In each curve in figure 3(a), v 0 is fixed, consequently, there exists an optimal Ω op at which S reaches its maximum value. On increasing v 0 , the position (Ω op ) of the peak shifts to large Ω and the height of the peak first increases and reaches its maximum, and then decreases. Figure 3(b) shows that the optimal Ω op is linearly related to v 0 , with a slope of 0.34. This means that the optimal radius R op (corresponding to the maximal S) is 0.34 at Φ = 0.5.
To study the dependence of the segregation coefficient S on Ω and v 0 in further detail, we have plotted the phase diagram of the binary mixture in the Ω − v 0 representation at Φ = 0.5 in figure 4. When either Ω < 1.0 or Ω > 6.9, the segregation coefficient S is small and the binary mixture remains mixed. When v 0 is   not large (e.g. v 0 = 2.0), the activity of the particles is too low to cause MIPS and the particles remain mixed.
When v 0 is very large, the radius of the circular motion becomes larger than the size of the system, making the chirality of the particles negligible. Therefore, for a given Ω, there exists a finite range of v 0 in which the mixed particles can be separated. As v 0 increases from 5.0, the demixing zone gradually expands. It is worth noting that the phase diagram does not change substantially with changes in the packing fraction Φ. Figure 5 shows the typical snapshots of the binary system for different Φ at v 0 = 10.0 and Ω = 3.0. At lower packing fractions (e.g. Φ = 0.4), MIPS does not occur, so the alignment between particles is weak or absent (as shown in figure 6), resulting in the absence of particle clustering. In this case, even if two types of  particles rotate in opposite directions, they cannot be separated (shown in figure 5(a)). But when the packing fraction reaches a level sufficient for MIPS (e.g. Φ = 0.5 and 0.6), the local alignment between particles becomes stronger and the correlation length increases (shown in shown in figure 6), as a result, CW particles will clump together in one cluster and CCW particles will clump together in the other cluster. The particles within each cluster move together, allowing the binary mixture to be separated, as depicted in figures 5(b) and (c). However, at higher densities (e.g. Φ > 0.7), MIPS gradually disappears as the system tends to become almost crystallized. At this stage, particles of the same type still accumulate together, but particles of different types also accumulate together, so the mixture is not separated (as shown in figure 5(d)). Therefore, choosing an appropriate packing fraction is crucial to achieve successful separation of the mixture.
The separation coefficient S as a function of the packing fraction Φ is shown in figure 7 for different cases. It is found that the binary mixture is always mixed if Φ < 0.45 or Φ > 0.7. This can be explained by the fact that when Φ < 0.45, MIPS does not occur, preventing the particles from accumulating into clusters; thus, the binary mixture remains mixed. When Φ > 0.7, the particle density is very high, and even crystallization occurs; all the particles then aggregate into a single large cluster, within which different types of particles mix and thus the mixture does not separate. In the range 0.45 < Φ < 0.7, however, MIPS can occur and particles do not crystallize so that the binary mixture can be separated. The segregation coefficient S is a peaked function of Φ, and the optimal Φ can maximize the separation of the mixture. Increasing v 0 causes the peak position to shift to lower Φ values, and the height of the peak first increases and then decreases (shown in figure 7(a)). Similarly, increasing Ω causes the peak position to shift to higher Φ values while the height of the peak remains subject to the same behavior (shown in figure 7(b)).
We plot the phase diagram of the binary mixture in the Φ − v 0 representation (shown in figure 8(a)) at Ω = 3.0 and the Φ − Ω representation (shown in figure 8(b)) at v 0 = 10.0. Figure 8(a) shows that the binary  mixture can always be separated for 5 < v 0 < 15, but with the increase of v 0 , the demixing zone first expands and then decreases. The optimal Φ (corresponding to the maximal S) decreases with the increase of v 0 . The phase diagram in the Φ − Ω representation shows that the mixture may be separated only when 1.0 < Ω < 8.0 and 0.45 < Φ < 0.7. The demixing zone expands when Φ increases from 0.45 and the optimal Φ increases with Ω. Note that these phase diagrams do not change qualitatively when the other parameters are varied.
Finally, we investigate the effect of the system size on the separation behavior of the binary mixture. In figure 9(a), we plot the segregation coefficient S vs the radius R = v 0 /Ω of the circular motion for different system size L = √ Nπσ 2 /(4Φ) at Φ = 0.5 and v 0 = 10.0. Figure 9(b) shows the dependence of the peak position on the system size L. It is evident that the separation behavior of the binary mixture depends strongly on the system size, with the segregation coefficient S taking on a peaked functional dependence on the radius R. The position of the peak shifts to larger R values as L increases. The optimal radius R op (which corresponds to the maximal S value) is found to depend quasilinearly on the system size, R op ∼ L. The position of the peak shifts to large R as L increases. In addition, when L increases to a larger size, the height of the peak drops drastically. This can be understood with recognition that, as the order caused by the hidden velocity alignment is local, it is impossible for particles to move together in a globally-ordered fashion in a large-sized system. The maximum cluster (the area size defined as A c ) formed by particles is dependent on the radius R of the circular motion, whereby larger radii allow for larger clusters to be formed. In a large-sized system (L 2 ≫ A c ), similar particles can only join in a multitude of small clusters, thus the system is mixed. Conversely, when A c matches L 2 , a perfect segregation of similar particles is seen since they are all largely contained within a single cluster. As a result, the optimal radius R op increases with L.

The effects of rotational and translational diffusion
In real active particle systems, noise cannot be ignored, and noise often has important effects on the dynamic behavior of the system. We need to investigate the effects of translational and rotational diffusion on particle separation. In this section, we explore the segregation dynamics of the binary mixture by varying rotational diffusion D r , translational diffusion D 0 , and the angular velocity Ω at Φ = 0.5 and v 0 = 10.0. Figure 10(a) displays the segregation coefficient S as a function of Ω for different D r at D 0 = 0. It is observed that, similar to the case when noise is absent (D r = 0), the segregation coefficient S exhibits a peaked function of Ω. As D r increases, the position of the peak shifts to smaller values of Ω. This can be explained as follows: from the previous analysis, the peak value corresponds to the optimal radius R op of circular motion. When D r = 0, R op = v 0 /Ω op . As D r increases from zero, an effective torque Ω Dr is produced, which shifts the optimal radius to R op = v 0 /(Ω op + Ω Dr ). Therefore, Ω op decreases with increasing D r .
In addition, the separation coefficient S, exhibits an intriguing behavior when D r increases from 0. Initially, S increases until reaching its maximum value, after which it rapidly decreases. This suggests that rotational diffusion does not always impede the separation process. To illustrate this phenomenon, we consider the case of Ω = 2.0 shown in figure 10(a). For successful separation of a mixture, particles of the same kind must aggregate, while those of different types should repel one another. This implies that the alignment interaction among particles does not consistently enhance the separation. During MIPS process, particles tend to form clusters due to the effective interparticle attraction. When D r = 0, the effective attractive force between particles is considerable, which prevents some heterotypic particles from repelling each other effectively. As demonstrated in figure 11(a), clusters comprising one particle type mix with a substantial number of particles of another type, hindering the separation process. However, as D r increases, the effective attractive force between particles weakens, enabling more heterotypic particles within clusters to escape. As displayed in figure 11(b), the number of heterotypic particles in the clusters decreases, contributing to a more effective separation. Nevertheless, when D r surpasses a certain threshold, the effective attractive interaction becomes significantly weaker. Consequently, large clusters dissociate into smaller ones, as depicted in figure 11(c). Ultimately, only small clusters form, as shown in figure 11(d), rendering the separation process ineffective. This behavior is further evidenced in figure 10(b), where S peaks as a function of D r . Thus, an optimal diffusion coefficient can promote the separation of the mixture.
The effect of translational diffusion on the mixture separation is shown in figure 12 at D r = 0 and Ω = 3.0. Similar to figure 10(b), the segregation coefficient S is a peaked function of translational diffusion D 0 . When D 0 is very large, the system's dynamic behavior is dominated by translational diffusion, and the dynamic differences between the two types of particles are eliminated, preventing their separation. When D 0 is not too large, translational diffusion may appropriately disrupt the tight clusters formed by particles, promoting their separation. Hence, translational diffusion does not always hinder the separation of mixtures.

The effects of polar velocity alignment interactions
In the first section, we examine the instance where MIPS causes hidden velocity alignment. To carry out a comprehensive comparison with explicit polar velocity alignment, we must take into account polar alignment interactions [38][39][40] in addition to the excluded volume interactions, as detailed in equation (2) and where the sum runs over the neighbors within a radius of r 0 around particle i. Torques resulting from polar velocity alignment interactions are incorporated in equation (6) with the coupling constant g ⩾ 0 determining their strength. Our main aim is to compare explicit polar velocity alignment with hidden velocity alignment caused by MIPS with a view to separating the mixture. To compare the effects of two velocity alignment methods on particle separation, we studied two representative cases. The first case has v 0 = 10.0 and Ω = 3.0, resulting in a system that exhibits MIPS. The second case has v 0 = 1.0 and Ω = 0.3, resulting in a system that does not exhibit MIPS. Figure 13 shows the segregation coefficient S as a function of the polar alignment strength g for different cases. Specifically, in the case where v 0 = 10.0 and Ω = 3.0 (MIPS), we observe that as g increases gradually from 0, S first decreases, reaches a minimum, then increases to a maximum, decreases again, and finally approaches zero. This can be explained as follows. When MIPS dominates the particle separation (g < 0.01), increasing g will break MIPS and hinder the separation process instead. However, when explicit polar velocity alignment dominates (g > 0.1), there exists an optimal value of g that maximizes the value of the segregation coefficient S. In contrast, when g exceeds a certain threshold, all particles form a large and tightly-clustered group, making it impossible to separate the two types of particles. For the case where v 0 = 1.0 and Ω = 0.3, MIPS is absent, and the mixture cannot be separated at g = 0. In this cases, particle separation is entirely caused by explicit polar velocity alignment, and S is a peaked function of g. By comparing these two cases, we can conclude that both explicit polar velocity alignment and hidden velocity alignment caused by MIPS can lead to the separation of mixtures, and these two types of alignment actions are competitive with each other. Therefore, hidden velocity alignment caused by MIPS has a different role from explicit polar velocity alignment in the particle separation process.
In the case of hidden velocity alignment in MIPS, the separation behavior of the system is strongly influenced by its size. This results from the fact that the largest particle cluster is determined by the radius R of the circular motion. On the other hand, when considering the polar alignment interaction between particles (as observed in the Vicsek type interaction [58]), the polar order is global and not dependent on both R and L. Thus, the maximum cluster size formed by the particles remains unaffected. Under strong polarity alignment, particles of the same species gather to create a large cluster. Consequently, the separation behavior in this scenario is not influenced by the size of the system. When examining both implicit and explicit velocity alignment at once, as depicted in equation (6), the impact of system size on separation behavior is determined by which alignment effect dominates the system's dynamics. If the implicit velocity alignment is predominant, then the separation behavior is size-dependent. However, under polar alignment predominance, the system size has no bearing on the separation behavior.

Conclusion and outlook
In this paper, we have conducted numerical studies on the separation behavior of a binary mixture consisting of CCW and CW particles in a two-dimensional box with periodic boundary conditions. Our results show that the mixed chiral particles can spontaneously demix without any external manipulations due to the presence of the necessary condition of MIPS. This phenomenon occurs because the hidden velocity alignment in MIPS results in the accumulation of particles into different clusters, leading to the segregation of the binary mixture. We found that the occurrence of MIPS is dependent on the packing fraction, with a disappearance of MIPS at Φ < 0.45 or Φ > 0.7, which results in a mixed binary mixture. We also discovered an optimum range for the angular velocity or packing fraction at which the segregation coefficient reaches its maximum value. Interestingly, we found that translational or rotational diffusion sometimes facilitates the separation of mixtures under certain conditions. Furthermore, we noticed that the optimal radius R op depends quasi-linearly on the system size L, as the order caused by the hidden velocity alignment in MIPS is local. Our comparison between particle separation caused by MIPS and explicit polar velocity alignment showed that the former mechanism leads to a novel method of separation.
Our work on the spontaneous demixing of mixed chiral particles does not require any external or internal manipulation, unlike previous studies [33][34][35][36][37][38][39][40][41][42][43][44][45][46] that needed special conditions. Our findings provide valuable insights into the fundamental demixing mechanisms of chiral active mixtures, which have practical implications in understanding and manipulating biological and synthetic micro-swimmers. Additionally, the novelty of our results may spawn further research on the physics of chiral active matter.