Active matter refers to a variety of systems ranging from schools of fish on a macroscale to swarms of bacteria on a microscopic scale.^[1] The common feature of these systems is that the constituents locally convert energy into motion. At a high density of constituent units, these systems can be viewed as a living material with the ability to adapt to stimuli, heal itself, like a biological tissue.^[2] The far-from-equilibrium feature of these systems leads to rich phenomena such as giant fluctuation,^[3–5] phase separation,^[6] and super-diffusion.^[7–9]

It was shown that active systems consisting of self-propelled particles interacting solely via repulsive interaction phase separate into a solid-like phase and a gas-like phase at very low area fractions in two dimensions.^[10] This motility induced phase separation (MIPS) has been observed in many simulations of self-propelled particles^[11,12] and also realized experimentally in a system of colloids.^[13] However, it is very challenging to reach a high packing fraction in experiments of pure self-propelled particles to investigate the phase-separation behaviors. Most experiments have been carried out with highly diluted suspension of self-propelled particles.^[14] Thus, numerical experiments were performed to study the complex phase behavior of dense self-propelled systems.

Recently, the mixtures of self-propelled particles (SPPs) and passive particles or chains in two dimensions have been intensively studied.^[15–17] The mixtures provide an additional degree of control for performing specific functions. For example, self-propelled particles lead to the globule–stretch transition of a self-attracting chain due to shear-induced stretching at a low rotational diffusion of SPPs or collision-induced melting at a high rate of SPPs.^[18] The mixtures of monodisperse repulsive active Brownian particles and their passive counterparts also display a MIPS behavior, similar to the monodisperse active system, except the existence of enhanced fluctuations and frequent fission and fusion of clusters.^[19] The big passive athermal particles display an effective attraction in the bath of small SPPs, which leads to a true phase separation for large size rations and area fraction of SPPs.^[17] Recent experiments and simulations also focused on the passive tracers with various shapes such as ellipsoid,^[20] chevrons,^[21] and wedge^[22] in the active suspensions, and found the existence of enhanced diffusion in the translational and rotational degrees. Although there are lots of works on monodisperse SPPs and mixtures of spherical passive particles and SPPs at high area fractions,^{[15,19,23–25]} the mixture of rod-like particles and SPPs has not been investigated so far.^[26]

In this paper, we computationally study a binary active mixture of rod-like particles and SPPs with total area fraction φ₀ = 0.6, using the active Brownian particle model. Here we focus on the low ratio of SPPs with the area fraction φ_a ∼ 0.015 (i.e., the area fraction of the passive particles is 0.585), which is easy to be realized in colloidal systems.^[27] Meanwhile, we also study the mixture of monodisperse SPPs and passive particles with the same fraction and the same ratio of SPPs. We address the following questions. What is the effect of the self-propelled force on the structure and dynamics of the two systems? And what is the structural difference between the two systems? We find both systems exhibit phase separation and super-diffusion as the self-propelled force is turned on, and the phase separation disappears at a large self-propelled force. Structurally, the rod-like particles accumulate loosely compared with the spherical particles at high-density regions due to the shape anisotropy.

The structure of this paper is as follows. We first outline our model and computational methods for both binary-mixture systems. Then, we describe our results and give necessary discussion. Finally, we provide the conclusion and outlook.

Our dimer system consists of a binary mixture of N_r rod-like particles (or N_p = 2N_r passive beads for the monomer system) and N_a SPPs moving on a two-dimensional frictional substrate (Fig. 1). Each SPP has its inherent orientation represented by a unit vector, n_i(θ_i) = (cosθ_i, sinθ_i), where θ_i is the angle between SPP’s orientation and the reference X-axis. This angle is subjected to rotational diffusion, as described by the rotational Brownian equation , where D_r is the rotational diffusivity and ξ_γ is the Gaussian white noise with unit variance.^[28]

Fig. 1.

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Fig. 1. Schematic of two systems, i.e., a monomer system consisting of monodisperse particles (blue) and self-propelled particles (SPPs) and a dimer system including rod-like particles and SPPs (red). The rod-like particle is composed of two monodisperse spheres connected by a stiff harmonic spring. The arrow indicates the orientation of the SPP which rotates under the thermal fluctuation. The eye-guiding dot line is drawn to separate the two systems.

To mimic the passive rod-like particles, two beads are connected by a stiff harmonic spring via a harmonic potential , where k = 10000k_BT/σ² is the spring constant, r is the center-to-center distance between the two bonded beads, and r₀ = 1.0 denotes the equilibrium distance. Here, k_BT is used as the energy unit, and σ is the length unit. For the monomer system, the passive particle is similar to the SPPs without inherent orientation.

The motion of each bead with constant mass m is governed by Newton’s equation^[29]

The pairwise interaction of the model is based on the dissipative particle dynamics (DPD) method^[30]

The pair interaction between any two beads in the system is modeled with a purely repulsive Weeks–Chandler–Andersen (WCA) potential

All simulations have been performed in a square box of size 100σ × 100σ with periodic boundary conditions using the home-modified LAMMPS software.^[31] The dimer system consists of N_r = 3700 rod-like particles and N_a = 200 SPPs. The monomer system consists of N_p = 7400 passive beads and N_a = 200 SPPs, insuring that the area fraction of the active particles is φ_a ∼ 0.015. We use m, σ, and k_BT as the fundamental units for mass, length, and thermal energy, respectively. The simulations parameters are chosen as m = 1, σ = 1, k_BT = 1, γ_dpd = 5.0, and D_r = 10^–3. We set γ_S = 0.1 for F_a ≠ 0. But we technically set γ_S = 0 when the active force is turned off with the aim of getting enough samples of typical structures. The dimensionless time unit is . The integration time step is set to Δ t = 10^–3 τ for small self-propelled force and Δ t = 5 – 10^–4 τ for large self-propelled force to ensure stable simulation results. In all simulations, the system typically runs up to 10⁷ steps.

Figure 2 shows the typical structures of the monomer and dimer systems at self-propelled forces F_a = 0, 1, 40. At F_a = 0, the system is in the thermodynamic equilibrium, which yields a homogeneous particle distribution. Afterwards the self-propelled force is turned on so that the active particles start performing active motion and colliding with nearby passive particles. Such collisions cause the creation of depleted regions behind the SPPs and the compression of the passive particles. For the monomer system, this is similar to what is observed in the recent experiment.^[27] For both systems, the SPPs result in the formation and compression of the clusters of passive particles at F_a = 1 (also see the supplemental movies). Thus, the phase separation characterized by the high-density and low-density regions forms. For the monomer system, the passive particles show a crystal-like cluster surrounded by many SPPs at the high-density region. The crystal-like cluster forms and vanishes with the evolution of simulation time. The crystal-like cluster melts from the boundary due to the continual collision with surrounding SPPs, which is in good agreement with a recent numerical simulation.^[32] For the dimer system, the degree of crystallization is very weak due to the rod-like shape of the passive particles. The distribution of SPPs is more random than that in the monomer system, in which the SPPs localize at the boundaries of the clusters. At F_a = 40, the distribution of particles in both systems becomes homogenous again. The phenomenon not obtained by the experiment and simulation^[27] may be due to the low velocity of the colloidal particles. The explanation is as follows: the effective temperature (data not shown) of the system increases with the active force of SPPs due to frequent collisions, the depleted region can be easily filled by the high-velocity passive particles. Besides, the melting of the small clusters is very fast even though they might emerge transiently. This is why the homogenous state with enhanced area variations is observed.

Fig. 2.

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Fig. 2. Snapshots of the monomer and dime systems at various self-propelled forces. SPPs are represented as red disks and passive particles including passive spheres and rod-like particles are represented as blue disks. The inset in the upper-right corner is the corresponding Voronoi cells for part of the monomer system at Fa = 1. The Voronoi area is the area of each polygon with blue edge in the inset. The inset in the lower-left corner is the enlarging crystal-like cluster for part of the monomer system at Fa = 1.

There are three mechanisms of phase separation in binary mixture systems with purely repulsive interaction. One is the depletion-induced phase separation. The depletion effect induces an attractive force between large particles in the mixture because of the unbalanced osmotic pressure exerted on them by the surrounding small particles.^[17] Active particles can also induce effective attraction between large passive colloidal particles, known as the active depletion.^[33] Under these conditions, the range and strength of the effective attraction are crucially dependent on the shape and size of the passive particle. It was also found that the eccentric active particles can push passive spheres to form a large dense dynamic cluster, and the mixture of the passive spheres and eccentric active particles undergoes a dynamic de-mixing transition due to the entropy-driven depletion.^[34] The second one is the diffusivity-induced phase separation in binary mixtures of monodisperse particles. An effective attractive interaction between the low diffusive particles results from the cage-effect by the surrounding high diffusive particles.^[35] The third one is the motility-induced phase separation^[36] and active–passive segregation.^[19] The active–passive mixture shows enhanced fluctuations, with frequent fission and fusion of clusters. The clusters are not homogeneous, but are predominantly active at the periphery and passive in their interior. Our mechanism is similar to the MIPS, which is critically dependent on Péclet number (Pe) of the active particles. Here , where is the swim velocity of an isolated particle and is the translational diffusion coefficient. It could be estimated that Pe ranges from ∼ 15 to ∼ 280 for the separation in our systems.

To qualitatively compare the structures of the two systems with the increase in self-propelled force, we first calculate the Voronoi area, S, of each bead (here we use “bead” to distinguish “particle”, because a rod-like particle consists of two beads). Figure 3 shows the deviation of the bead area, 〈 δ S² ⟩ = 〈 (S – S₀)²〉 (here S₀ is equal to the box area over bead number) as well as the average area, 〈S〉, of the passive particles and active particles. It can be seen from Fig. 3(a) that, for the homogenous equilibrium state (F_a = 0), the deviation is very small (< 0.02). When the active force is turned on, the deviation increases to 0.11, corresponding to the phase separation of the two systems. Then the deviation decreases with the active force of SPPs, implying that the system gradually turns to a homogeneous state. Interestingly, the deviation shows a slight increase with a further increase in active force in both cases. However, the minimal value of 〈 δS²〉 of the active system is larger than that in the equilibrium state. This implies that the systems with a slightly large fluctuation at the high active force of SPPs are different from the thermo-equilibrium system.

Fig. 3.

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	Fig. 3. (a) The deviation of Voronoi area as a function of Fa. The average Voronoi area of (b) passive beads and (c) active particles as a function of Fa for both systems.

The average area of the passive beads increases with the increase in active force, while that of the active particles decreases with the increase in active force (Figs. 3(b) and 3(c)). Both of them plateau to a nearly constant value for high active force. The difference is that the value of 〈S〉 for the passive beads of the monomer system is larger than that of the dimer system. The discrepancy results from the shape difference of spherical particles and rod-like particles. As expected, the average area of the active particle shows an opposite trend due to the area conservation of the systems.

To get the quantitative difference of the structure of the binary mixture, we calculate the pair correlation functions (PCFs), g(r), between the particle pairs. It is a measure of the probability of finding a bead around the reference one. We present the PCFs between passive particles in Fig. 4 for the monomer, g(r)_mm, and dimer, g(r)_dd, systems, and g(r) between active particles at the corresponding self-propelled forces. It can be found that there are many peaks of g(r)_mm due to the dense mixture we used at the equilibrium state, as shown in Fig. 4(a), indicating a layer-like structure in the system. When the active force is turned on, the peaks become higher, especially, the first peak and the appearance of many new peaks at large radial distances, r, relative to the equilibrium case, implying the compression of the passive particles, which is in good agreement with the typical snapshot of Fig. 1. As the active force further increases, the peak drops and disappears at large r (as shown in Fig. 4(a) at F_a = 20 and F_a = 80). It should be noted that the PCF can not give the information of phase separation. The behavior of PCFs for the dimer system (Fig. 4(b)) is similar to that for the monomer system. The differences are that: 1) the first peak for each F_a shows a little drop because of the spring connection of the two beads of the rod-like particles; 2) the peaks vanish at a moderately shorter distance, indicating a slightly loose packing of rod-like particles compared with that of the monomer system.

Fig. 4.

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	Fig. 4. The PCF of passive particles for (a) the monomer system and (b) the dimer system. (c) The PCF of active particles for both systems at various Fa.

To further clarify the difference between the two systems, we also present g(r) of active particles in both systems for various active forces in Fig. 4(c). One can find that the g(r) peaks of active particles for the dimer system at F_a = 1.0 are lower than those for the monomer system. Also, the second and third peaks show a slight shift to large r value. This indicates that the layer structure of the active particles in the dimer system is looser than that in the monomer system. At F_a = 20 and 80, the behavior of g(r) shows little difference. And the peak is very low compared to that at F_a = 1.0. This implies that a homogenous structure reappears.

To study the correlation between velocity and area of the particles, we give the joint probability of velocity and Voronoi area of the active particles in Fig. 5 for the two systems at F_a = 1.0, 5.0, 20.0, respectively. Here the velocity is the average velocity of particles per τ. It can be seen that the distribution of the particle area moderately expands with the increase in the self-propelled force. The area of the active particles in the monomer system is between 1.1 and 1.6. While it is between 1.1 and 1.75 in the dimer system. Furthermore, the particle velocity increases with the increase in self-propelled force for both systems. Moreover, the joint probability also shows that the higher the particle’s velocity is, the larger the area is, especially, for the case of the small self-propelled force. This indicates that here exists a positive correlation between the Voronoi area and velocity of the particles. Intuitively, the high velocity of SPPs will keep a large depleted region under the same response time (or average velocity) of the passive particles.

Fig. 5.

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	Fig. 5. The joint probability of velocity and Voronoi area of active particle for monomer (a) and dimer (b) systems at Fa = 1.0 (left), 5.0 (middle), 20.0 (right).

To obtain the influence of SPP on the dynamics of the passive particles, we measure the mean square displacements (MSD) of the passive particles, 〈(r(t))²〉 = 〈(r(t) – r(0))²〉, at various F_a for both systems. We also calculate the scaling exponent β(t) = d[log(〈(Δr(t))²〉)]/d[log(t)] varying with time, as shown in Fig. 6. For normal Brownian diffusion, β(t) = 1. It can be seen that β(t) = 1 at F_a = 0 for both systems. At F_a = 1.0, the passive particles exhibit a super-diffusion behavior at a short time scale, then gradually turn to normal diffusion at a long time scale. As shown in Fig. 6(b), β(t) decays quickly from ∼ 1.8 to ∼ 1.0 with the increase in time. At F_a = 20, the passive particles sustain the super-diffusive behavior in the window of our simulation time. The β(t) firstly decreases and then increases with time in our simulation time scale, although the MSD should recover the normal diffusion at a very long time. The similar behavior also exists for the copolymer chain immersed in the bath of SPPs.^[37] The mechanism of the nonmonotonic behavior of β(t) is not clear, possibly due to the formation and melting of a small cluster of passive particles in the system. This result indicates that a very dilute active particle can result in the super-diffusion of the passive particles.

Fig. 6.

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	Fig. 6. Log–log plots of (a) the mean square displacement of passive particles for the monomer and dimer systems. The dotted lines show slopes of 2.0 and 1.0. (b) The variations of the scaling exponents β(t) for both systems at the indicated self-propelled forces.

The study of how SPPs behave in crowded environments resembles realistic situations under which SPPs may be employed, e.g., as drug delivery systems. We performed molecular dynamics simulations with DPD thermostat to study two kinds of binary-mixtures in two dimensions: one is the mixture of passive and self-propelled spheres, the other is the mixture of rod-like particles and self-propelled spheres. Our results show that a very small number of active particles strongly influence the distribution and the dynamics of the dense mixtures. We find that, at a small active force, the high-density and low-density regions appear in both systems, indicating an emergence of phase separation. Then it vanishes at higher active forces due to the velocity improvement induced by frequent collision. Structurally, the rod-like particles accumulate loosely due to the shape anisotropy, compared with the spherical particles at the high-density region. In addition, we find the existence of positive correlation between Voronoi area and velocity of particles in both systems. Moreover, the passive particles show a super-diffusive behavior when the self-propelled force is turned on in spite of the low fraction of the active particles.

Here, we focus only on the dense systems with a low fraction of SPPs. In the future, the fraction effect of SPPs will be studied and the phase diagram might be given for the rod-like particle system.^[38] Besides, the length of the rod-like particles is another interesting parameter that should be studied in further work.^[39]