Microscopic origin and macroscopic implications of lane formation in mixtures of oppositely driven particles

Colloidal particles of two types, driven in opposite directions, can segregate into lanes [Vissers et al., Soft Matter 7, 2352 (2011)]. This phenomenon can be reproduced by two-dimensional Brownian dynamics simulations of model particles [Dzubiella et al., Phys. Rev. E 65, 021402 (2002)]. Here we use computer simulation to assess the generality of lane formation with respect to variation of particle type and dynamical protocol. We find that laning results from rectification of diffusion on the scale of a particle diameter: oppositely driven particles must, in the time taken to encounter each other in the direction of the drive, diffuse in the perpendicular direction by about one particle diameter. This geometric constraint implies that the diffusion constant of a particle, in the presence of those of the opposite type, grows approximately linearly with the Péclet number, a prediction confirmed by our numerics over a range of model parameters. Such environment-dependent diffusion is statistically similar to an effective interparticle attraction; consistent with this observation, we find that oppositely driven nonattractive colloids display features characteristic of the simplest model system possessing both interparticle attractions and persistent motion, the driven Ising lattice gas [Katz, Leibowitz, and Spohn, J. Stat. Phys. 34, 497 (1984)]. These features include long-ranged correlations in the disordered regime, a critical regime characterized by a change in slope of the particle current with the Péclet number, and fluctuations that grow with system size. By analogy, we suggest that lane formation in the driven colloid system is a phase transition in the macroscopic limit, but that macroscopic phase separation would not occur in finite time upon starting from disordered initial conditions.

Figure 1

Rectification of diffusion on the scale of a particle diameter underpins the laning transition. Red and blue particles moving persistently in opposite directions (left-right) must diffuse laterally (up-down) by about a particle diameter in the time taken to encounter each other. Driven rightward diffusion of the blue particle in the rest frame of the red particle (at center) shows the rectification of lateral diffusion that occurs if red and blue particles exclude volume (blue line); the large gray circle indicates the position of closest possible approach of the centers of red and blue particles. The green trajectory shows similar driven diffusive motion in the absence of volume exclusion. Both trajectories were generated using the dynamics described in Appendix pp1.

Figure 2

Steady-state dynamic regimes observed using our three numerical protocols for systems with 2500 particles, starting from disordered initial conditions at density $\rho =0.5$. The vertical axis is the Péclet number. The horizontal axis interpolates between the Langevin simulations of Protocol I (at “zero step size,” i.e., ${\mathrm{\Delta }}_x=0$) and the lattice-based simulations of Protocol III (${\mathrm{\Delta }}_x=1$), with the results of off-lattice hard-disk Monte Carlo simulations (Protocol II) shown for a range of intermediate step sizes ${\mathrm{\Delta }}_x$. Snapshots A, C, and D were obtained using Protocol II; B was obtained using Protocol I; and E was obtained using Protocol III. In the snapshots, the driven direction is left-right. The lines show the approximate boundaries between different steady-state behaviors: the black dashed line shows the boundary between the jammed and flowing states, and the solid red line shows the boundary between disordered and laned states (see Appendix pp1 for more details). Monte Carlo simulations reproduce the results of Langevin simulations if the basic step size of the former is small enough; otherwise, Monte Carlo and Langevin results differ qualitatively. Lattice-based Monte Carlo simulations jam for a Péclet number of order unity.

Figure 3

The lateral diffusion constant (perpendicular to the driven direction) as a function of Pe for a test particle in the presence of particles of the other type. We have normalized the diffusion constant by the bare value. The longitudinal component of diffusion behaves similarly; see Fig. 11. In these simulations, one blue particle is placed in a box of red particles, and the blue particle's diffusion constant is measured. This enhancement of diffusion with Pe of one particle in the presence of particles of the other type underpins the laning transition, seen for Langevin simulations and off-lattice Monte Carlo simulations with a sufficiently small basic step size. Off-lattice Monte Carlo protocols with larger step sizes do show diffusion enhancement with Pe, but tend to jam rather than to exhibit laning. On-lattice Monte Carlo protocols show no enhancement of diffusion constant with Péclet number, and they jam readily.

Figure 4

The enhancement of lateral diffusion as a function of Péclet number for one particle in the presence of particles of the opposite type (bottom panel) correlates approximately with an increase in an order parameter $\varphi$ for lane formation in equimolar mixtures (top panel). Both calculations used Langevin dynamics (Protocol I). The bottom panel comes from the steady-state simulations of Fig. 3, in which a test particle is placed in a simulation box containing only particles of the other type. The top panel comes from the equimolar red-blue mixture simulations of Fig. 2.

Figure 5

(a) Trajectories of the $y$ coordinate of a single blue particle placed in a periodic box of red particles present at density $\rho =0.5$ (see Fig. 3), for $\mathrm{Pe}=0$ and 160. Particles are driven in the $x$ direction. These trajectories show visually the enhanced diffusion in the presence of the drive. (b) Histograms of the lateral fluctuations for various Pe can be collapsed (c) by rescaling $\mathrm{\Delta }{y}_{\tau }$ by ${(a+b\mathrm{Pe})}^{-1/2}$, where $a$ and $b$ are constants, as suggested by Eq. (10). This collapse indicates that the simple physical argument that gives rise to Eq. (10) captures, in this parameter regime, the microscopic origin of enhanced diffusion.

Figure 6

Diagram used to demonstrate the statistical equivalence between hard particles with environment-dependent hopping rates $f$ and hard particles with environment-independent hopping rates and interactions of strength $k_{\mathrm{B}}Tlnf$.

Figure 7

Velocity correlation functions, Eq. (14), along (a) and against (b) the direction of the drive, measured in a simulation box of size $336\sigma \times 84\sigma$ at $\rho =0.5$. Long-range correlations in panel (a) are evident well below the critical Péclet number of 90. Above this value, the function plotted in (b) acquires an oscillatory structure, signaling the formation of lanes parallel to the driven direction.

Figure 8

The off-lattice model displays (a) a change of slope of particle current with Pe (the dashed black line shows a linear fit to the current values for Péclet numbers 50–85 in order to highlight the change in slope) and (b) system-spanning current fluctuations, similar to the behavior of the driven lattice gas at its critical point. $N$ indicates the number of particles present in the simulation box. Fluctuations of individual particle diffusion constants behave similarly (see Fig. 12).

Figure 9

The off-lattice model displays, in the two-phase region, a slow coarsening process similar to that seen in the driven lattice gas. Panel (a) shows a coarsening process starting from disordered initial conditions ($\mathrm{\Delta }t$ is the Langevin time step). Panel (b) shows a snapshot obtained by choosing as the initial condition two lanes, which persisted upon simulation.

Figure 10

Activity (scaled by the average activity for an isolated particle) and the laning order parameter $\varphi$ as a function of MC sweep for WCA particles at three different step sizes.

Figure 11

The longitudinal mean-squared displacement (in the driven direction) of a test particle in the presence of particles of the other type, normalized by the test particle's bare mean-squared displacement, as a function of Péclet number, for different dynamic protocols.

Figure 12

Lateral mean-squared displacement [here $\mathrm{\Delta }{y}_{\tau }=y\left(t\right)-y(t+\tau )]$ distributions broaden near criticality in a manner similar to distributions of particle currents; see Fig. 8.

Figure 13

Measured diffusion constants for a range of densities and Peclet numbers. The linear scaling of the lateral diffusion constant with Pe suggested by Eq. (10) is evident for a range of Pe and $\rho$ (top right panel), and it breaks down at a large packing fraction and a small Pe.