Microscopic field-theoretical approach for mixtures of active and passive particles

We consider a phase field crystal modeling approach for mixtures of interacting active and passive particles in two dimensions. The approach allows us to describe generic properties for such heterogeneous systems within a continuum model. We validate the approach by reproducing experimental results, as well as results obtained with agent-based simulations. The approach is valid for the whole spectrum from highly dilute suspensions of passive particles and interacting active particles in a dense background of passive particles. However, we concentrate only on the extreme cases, because for the situation with similar fractions of active and passive particles emerging structures are hard to analyze and experimental results are missing. We analyze in detail enhanced crystallization due to the presence of active particles, how collective migration is affected by a disordered environment, and laning states, which are globally nematic but polar within each lane.

Figure 1

(a), (b) Time series of a head-on collision between an active and a passive particle for low passive mobility $M_0^B=10$ (a) and high passive mobility $M_0^B=70$ (b). The active particle is shown (at different times) as a contour line of ${\psi }_A$, and its orientation is represented by an arrow. Blue (red) is used to show the particle before (after) the collision. The final position of the passive particle is shown as a black disk, whereas in (b) the transparent black disk represents the initial condition of the passive particle, and the black line its trajectory. The active particle bounces back after the collision (a) and transport the passive particle (b). The velocities along the $x$ direction of active and passive particles are shown as a function of time in panels (c) for $M_0^B=10$ and (d) for $M_0^B=70$; $M_0^A=70$ for both cases. The yellow region represents the approximate time of the collision.

Figure 2

Snapshots showing passive clusters for different total densities and fractions of active particles $\varphi$ and ${\eta }_A$ at time $t=0$ (first column) and at time $t=1000$ (second column). Particles with the same color belong to the same cluster, white disks represent passive particles not belonging to any cluster, and black disks are active particles. (a) $\varphi =0.5, {\eta }_A=0.05$, (b) $\varphi =0.6, {\eta }_A=0.05$ (c) $\varphi =0.7, {\eta }_A=0.05$, (d) $\varphi =0.8, {\eta }_A=0.15$. Other parameters are $M_0^A=M_0^B=50$.

Figure 3

Percentage of passive particles belonging to a cluster $X_f$ as a function of time for different total densities and fractions of active particles $\varphi$ and ${\eta }_A$. We observe that for $\varphi =0.5$ and $\varphi =0.6$ (top row), increasing the number of active particles leads to an increase of $X_f$, whereas the opposite is true for $\varphi =0.7$ and $\varphi =0.8$ (bottom row). Other parameters are $M_0^A=M_0^B=50$. Each curve has been obtained as the average of five different simulations starting with different initial conditions.

Figure 4

Maximum of the particle-averaged mean square displacement $\langle \mathrm{\Delta }{r}^2\left(t\right)\rangle$ of active particles moving in a binary mixture for different values of $\varphi$ and ${\eta }_A$. Active particles travel a longer distance when the passive particles have not crystallized, until the extreme case of $\varphi =0.8, {\eta }_A=0.05$ where the maximum mean square displacement is so low that active particles are basically trapped (see also Supplemental video SV1 in Ref. [50]). Other parameters are $M_0^A=M_0^B=50$. Each point has been obtained as the average of five different simulations started with different initial conditions.

Figure 5

(a) Average velocity ${\tilde{v}}_B$ of passive particles as a function of their mobility in an active bath with $\varphi =0.9,{\eta }_A=0.9$. ${\tilde{v}}_B$ increases almost linearly for small $M_0^B$ until it starts to saturate at around $M_0^B=70$. (b) Order parameter $\omega$ as a function of time for different mobility $M_0^B$. For small values of $M_0^B$ there is no collective migration, for intermediate values this state is reached quite fast, whereas for high mobility the transient phase to reach collective migration increases. We choose to fix $M_0^B=70$ for the analysis in Fig. 6. $M_0^A=100$ for both cases. The data have been obtained as the average of 10 different simulations started with different initial conditions.

Figure 6

Order parameter $\omega$ as a function of time for different values of $\varphi$ and ${\eta }_A$. The purple curve corresponds to the case ${\eta }_A=1$, i.e., no passive particles present. We see that in all other cases the state of collective migration is reached later (longer transient phase) or not reached at all, especially for lower ${\eta }_A$ (red curves). Other parameters are $M_0^A=100$ and $M_0^B=70$. The data have been obtained as the average of 10 different simulations begun with different initial conditions.

Figure 7

The color code corresponds to the orientation of the single-particle velocity, and black disks represent passive particles. (a) Snapshot of a laning state, with two macroregions of active particles having exactly opposite orientation. This state can last for a long time because of the presence of passive particles at the boundary between the two regions (see also Supplemental video SV2 in Ref. [50]). (b) Another laning state. Here less passive particles are accumulated at the boundary between the moving active regions. This situation is less stable, and, at a later time (c), active particles all move in the same direction, whereas passive ones form chains, which persist over longer periods (see also Supplemental video SV3 in Ref. [50]). (d) Passive particles forming clusters in an active bath. (a)–(c) Regime $\varphi =0.8, {\eta }_A=0.9$, (d) regime $\varphi =0.9, {\eta }_A=0.7$.