Dynamics of Magnus-dominated particle clusters, collisions, pinning, and ratchets

Motivated by the recent work in skyrmions and active chiral matter systems, we examine pairs and small clusters of repulsively interacting point particles in the limit where the dynamics is dominated by the Magnus force. We find that particles with the same Magnus force can form stable pairs, triples, and higher ordered clusters or exhibit chaotic motion. For mixtures of particles with opposite Magnus force, particle pairs can combine to form translating dipoles. Under an applied drive, particles with the same Magnus force translate; however, particles with different or opposite Magnus force exhibit a drive-dependent decoupling transition. When the particles interact with a repulsive obstacle, they can form localized orbits with depinning or unwinding transitions under an applied drive. We examine the interaction of these particles with clusters or lines of obstacles and find that the particles can become trapped in orbits that encircle multiple obstacles. Under an ac drive, we observe a series of ratchet effects, including ratchet reversals, for particles interacting with a line of obstacles due to the formation of commensurate orbits. Finally, in assemblies of particles with mixed Magnus forces of the same sign, we find that the particles with the largest Magnus force become localized in the center of the cluster, while for mixtures with opposite Magnus forces, the motion is dominated by transient local pairs or clusters, where the translating pairs can be regarded as a form of active matter.

Figure 1

The particle locations (dots) and trajectories (lines) for pairs of interacting particles. (a) When ${\alpha }_m^1={\alpha }_m^2=1.0$, the particles form a clockwise rotating bound pair. (b) For ${\alpha }_m^1=2.0$ and ${\alpha }_m^2=1.0$, the particles form nested orbits where the particle with the higher Magnus force is closer to the center. (c) A system with ${\alpha }_m^1={\alpha }_m^2=1.0$ in which a finite damping term ${\alpha }_d=0.1$ has also been added, causing the particles to spiral out gradually. (d) The ${\alpha }_m^1={\alpha }_m^2=1.0$ system from (a) with an additional drift force $F_D=0.075$ applied in the $x$ direction, causing the pair to translate in the negative $y$ direction.

Figure 2

The particle locations (dots) and trajectories (lines) for pairs of interacting particles. (a) When ${\alpha }_m^1=2.0$ and ${\alpha }_m^2=-2.0$, the particles form a dipole that translates in a fixed direction. (b) When ${\alpha }_m^1=1.65$ and ${\alpha }_m^2=-2.0$, the dipole moves in a circular orbit.

Figure 3

Dots: The measured dipole velocity $V_d$ vs ${\alpha }_m$ for the system in Fig. 2 with ${\alpha }_m^1={\alpha }_m$ and ${\alpha }_m^2=-{\alpha }_m$, where the initial distance between the particles is $R=2.0$. The solid blue line indicates $1/{\alpha }_m$ behavior. (b) $V_d$ vs $R$ for the same system with fixed ${\alpha }_m=2.0$.

Figure 4

The velocities $V_1$ (blue) and $V_2$ (red) of a pair of particles vs $F_D$ for a system with ${\alpha }_m^1=1.6$ and ${\alpha }_m^2=2.0$, showing a drive-induced decoupling transition near $F_D=0.15$.

Figure 5

The particle position (red dot) and trajectory (line) with the obstacle location (blue dot) for a single particle interacting with a stationary obstacle in the form of a permanently fixed particle. The particle has ${\alpha }_m=2.0$ and is initialized at a distance $R=1.5$ from the obstacle, and the applied drive is (a) $F_D=0.005$, (b) $F_D=0.015$, (c) $F_D=0.0165$, and (d) $F_D=0.025$.

Figure 6

$|\langle V_y\rangle |$, the absolute value of the average velocity in the $y$ direction of the particle from the system in Fig. 5, vs $F_D$, showing a depinning transition at $F_D=0.016$.

Figure 7

The particle position (red dot) and trajectory (lines) along with the obstacle location (blue dot) for the system from Fig. 5 with ${\alpha }_m=2.0$ and $R=1.5$ at $F_D=0.01$ where an additional damping term of ${\alpha }_d=0.01$ has been added to the dynamics. The particle gradually spirals away from the obstacle.

Figure 8

The particle positions (blue and orange dots) and trajectories (lines) along with the obstacle position (red dot). (a) Two particles trapped at an obstacle for ${\alpha }_m^1={\alpha }_m^2=2.0$ at $F_D=0.005$ and $R=1.5$. (b) The same system at $F_D=0.01$ where only one particle can be trapped. (c) The same system at $F_D=0.01$ in which the two particles are initially in a rotating pair that collides with the defect, which traps one of the particles. (c) Two particles trapped at an obstacle for ${\alpha }_m^1=2.0$ and ${\alpha }_m^2=-2.0$ at $F_D=0.005$ and $R=1.5$, where the Magnus forces of the particles have opposite signs.

Figure 9

The particle position (blue dot) and trajectory (line) along with the obstacle positions (red dots) for a particle with ${\alpha }_m=1.0$ under a drive interacting with an array of obstacles placed in a funnel configuration. (a) At $F_D=0.0$, the particle is bound to the funnel array and follows an orbit that encircles the obstacles in a counterclockwise direction. (b) For a negative $x$ direction drive of $F_D=0.01$, the particle moves in the positive $y$ direction and deviates around the obstacles. (c) For a drive of $F_D=0.01$ applied in the positive $x$ direction, the particle moves in the negative $y$ direction but does not become trapped by the funnel tip. (d) The same as panel (c) at $F_D=0.250$, where the particle passes through the funnel.

Figure 10

(a) The absolute velocity $|\langle V_y\rangle |$ vs $F_D$ for the system in Figs. 9 and 9 for motion in the negative $y$ direction (blue) and positive $y$ direction (pink). For either direction of drive, in Region I, the particle moves around the obstacles, and in Region II, the particle breaks through the funnel between the outer two obstacles. The dashed line at $F_D=0.95$ indicates a transition for the positive $y$ direction motion to the flow illustrated in Fig. 9. Changes in the breakthrough location are associated with small cusps in the velocity-force curve, and additional breakthrough cusps occur at higher drives (not shown). (b) $|\langle V_y\rangle |$ vs $F_D$ for the same system in the overdamped limit of ${\alpha }_m=0.0$ and ${\alpha }_d=1.0$. There is a finite depinning threshold for motion in the negative $y$ direction (blue) but not for motion in the positive $y$ direction (pink), creating a diode effect.

Figure 11

The particle position (blue dot) and trajectory (line) along with the obstacle positions (red dots) for a particle with ${\alpha }_m=0.0$ and ${\alpha }_d=1.0$ at $F_D=0.04$ in the overdamped limit of the system in Fig. 10. (a) Motion in the negative $y$ direction where the particle becomes trapped. (b) Motion in the positive $y$ direction where the particle moves around the obstacle array and does not become trapped.

Figure 12

The particle position (blue dot) and trajectory (line) along with the obstacle positions (red dots) for a particle with ${\alpha }_m=2.0$ moving toward a line of repulsive obstacles. (a) At $F_D=0.007$, the particle trajectory deviates into the positive $x$ direction as it approaches the line of obstacles. (b) At $F_D=0.07$, the $x$-direction deviation is smaller.

Figure 13

The particle position (blue dot) and trajectory (line) along with the obstacle positions (red dots) for a particle interacting with a line of obstacles while subjected to an ac drive in the $x$ and $y$ directions. (a) A particle with $A=B=0.05$, $\omega =0.00005$, and ${\alpha }_m=2.0$ placed at $R=12a$, where there is no directed motion. (b) The same as panel (a) but with the particle placed at $R=2a$, where now directed motion occurs in the positive $x$ direction. (c) The same as panel (b) but with $A=B=0.1$, where the directed motion is in the negative $x$ direction. (d) The same as panel (b) but with ${\alpha }_m=10$, where the directed motion is in the positive $x$ direction.

Figure 14

(a) $\langle V_x\rangle$ vs $A$ for the system in Fig. 13 showing a current reversal. (b) $\langle V_x\rangle$ vs ${\alpha }_m$ for the system in Fig. 13 with $A=0.05$, showing a current reversal.

Figure 15

The particle position (blue dot) and trajectory (line) along with the obstacle positions (red dots) for a particle interacting with a line of obstacles under an ac drive with $A=B$ and $\omega =0.00005$. (a) The system in Fig. 14 with $A=B=0.05$ at ${\alpha }_m=1.0$ showing translation in the negative $x$ direction. (b) A translating orbit with $A=B=0.025$ where the particle does not encircle any obstacles. (c) The same as in (b) but with the particle initially placed below the line of obstacles, which produces translation in the negative $x$ direction. (d) The system from Fig. 13 with only one direction of ac drive, achieved by setting $A=0.05$ and $B=0.0$. The ratchet effect operates at only half the velocity found for simultaneous $x$ and $y$ driving with $A=B=0.05$.

Figure 16

(a) $V_x$ vs time in simulation time steps for a particle with ${\alpha }_m=2.0$ interacting with a line of obstacles at an initial distance of $R=4a$ with ${\alpha }_d=0.005$ (black) and ${\alpha }_d=0.001$ (red). (b) $V_x$ vs time for the same system with ${\alpha }_m=2.0$ and ${\alpha }_d=0.005$ for particles initialized at $R=a$ (black), $2a$ (red), $4a$ (green), and $6a$ (blue).

Figure 17

The particle positions (dots) and trajectories (lines) for multiparticle systems with no drive. (a) $N=3$, ${\alpha }_m^1={\alpha }_m^2={\alpha }_m^3=1.0$ (blue). (b) $N=3$, ${\alpha }_m^1={\alpha }_m^3=2.0$ (blue), and ${\alpha }_m^2=1.0$ (purple). (c) $N=3$, ${\alpha }_m^1=1.0$ (blue), ${\alpha }_m^2=3.0$ (light purple), and ${\alpha }_m^3=2.0$ (dark purple). (d) $N=3$, ${\alpha }_m^1=1.0$ (blue), ${\alpha }_m^2=7.0$ (light blue), and ${\alpha }_m^3=2.0$ (purple). (e) $N=4$, ${\alpha }_m^1=1.0$ (dark blue), ${\alpha }_m^2=2.0$ (dark purple), ${\alpha }_m^3=3.0$ (medium purple), and ${\alpha }_m^4=4.0$ (light blue). (f) $N=4$, ${\alpha }_m^1={\alpha }_m^2={\alpha }_m^3={\alpha }_m^4=2.0$ (blue).

Figure 18

The particle positions (dots) and trajectories (lines) showing ringlike structures in multiparticle systems with strong variations in the Magnus forces. (a) $N=4$, ${\alpha }_m^1=7.0$ (light blue), ${\alpha }_m^2={\alpha }_m^3=2$ (purple), and ${\alpha }_m^4=1$ (dark blue). (b) $N=4$, ${\alpha }_m^1=10$ (light blue), ${\alpha }_m^2=1.5$ (dark purple), and ${\alpha }_m^3={\alpha }_m^4=2$ (light purple). (c) $N=4$, ${\alpha }_m^1={\alpha }_m^2=10$ (light blue), and ${\alpha }_m^3={\alpha }_m^4=3$ (light purple), showing a dumbbell structure. (d) $N=4$, ${\alpha }_m^1={\alpha }_m^2={\alpha }_m^3=7$ (light blue), and ${\alpha }^4=2$ (purple).

Figure 19

(a, b) The particle positions (dots) and trajectories (lines) for two particles initialized at opposite ends of the sample under a driving force of $F_D=0.075$. (a) The first portion of the collision for ${\alpha }_m^1=2.0$ and ${\alpha }_m^2=-2.0$. (b) Continuation of the motion in (a) after the particles have passed one another. (c) Collision for ${\alpha }_m^1=-2.0$ and ${\alpha }_m^2=1.85$. (d) Collision for ${\alpha }_m^1=2.0$ and ${\alpha }_m^2=-1.0$.

Figure 20

The particle positions (dots) and trajectories (lines) in systems with mixed Magnus force sign and no drive. (a) A closed orbit at $N=3$, ${\alpha }_m^1=1.0$ (purple), ${\alpha }_m^2=-1.1$ (orange), and ${\alpha }_m^3=0.85$ (blue). (b) A translating dipole at $N=5$, ${\alpha }_m^1={\alpha }_m^2={\alpha }_m^3={\alpha }_m^4=2.0$ (blue), and ${\alpha }_m^5=-2.0$ (red). The two particles that are paired into the dipole are at the bottom and top of the image due to the periodic boundary conditions. (c) At $N=5$, ${\alpha }_m^1={\alpha }_m^3={\alpha }_m^5=2.0$ (blue), and ${\alpha }_m^2={\alpha }_m^4=-2.0$ (red), there are two intermittently forming pairs of dipoles. (d) Circular dipole motion at $N=5$, ${\alpha }_m^1={\alpha }_m^2={\alpha }_m^5=2.0$ (blue), and ${\alpha }_m^3={\alpha }_m^4=-1.5$ (red).

Figure 21

The particle positions (dots) and trajectories (lines) in systems with mixed Magnus force sign and no drive. (a) For $N=4$, ${\alpha }_m^1={\alpha }_m^3=2.0$ (blue), and ${\alpha }_m^2={\alpha }_m^4=-4.0$ (red), the particles form a translating cluster. (b) For $N=4$, ${\alpha }_m^1=1.0$ (blue), ${\alpha }_m^2={\alpha }_m^4=-2$ (red), and ${\alpha }_m^3=2$ (purple), the cluster moves in a circle.

Figure 22

The velocity component $V_x$ vs time in simulation time steps for one of the particles in Fig. 20 showing jumps between zero velocity when the particle is not part of a dipole and finite velocity when the particle is part of a translating dipole.