Active Microrheology, Hall Effect, and Jamming in Chiral Fluids

TL;DR

This study investigates the motion of a probe in a chiral fluid of circular swimmers, revealing nonmonotonic velocity behavior, a resonance-driven Hall angle, and jamming transitions influenced by swimmer activity and density.

Contribution

It introduces the first analysis of Hall effect and jamming phenomena in chiral active fluids with circularly swimming particles.

Findings

Probe exhibits both longitudinal and transverse motion with a Hall angle.

Hall angle peaks at resonance between swimmer frequency and probe motion.

Jamming occurs at high swimmer frequencies and high densities.

Abstract

We examine the motion of a probe particle driven through a chiral fluid composed of circularly swimming disks. We find that the probe particle travels in both the longitudinal direction, parallel to the driving force, and in the transverse direction, perpendicular to the driving force, giving rise to a Hall angle. Under constant driving force, we show that the probe particle velocity in both the longitudinal and transverse directions exhibits nonmonotonic behavior as a function of the activity of the circle swimmers. The Hall angle is maximized when a resonance occurs between the frequency of the chiral disks and the motion of the probe particle. As the density of the chiral fluid increases, the Hall angle gradually decreases before reaching zero when the system enters a jammed state. We show that the onset of jamming depends on the chiral particle swimming frequency, with a fluid state appearing at low frequencies and a jammed solid occurring at high frequencies.

Equations

$$\eta \frac{d{\mathbf{r}}_i}{dt}=\sum _{j\ne i}^N{\mathbf{F}}_{pp}^{ij}+{\mathbf{F}}_{\mathrm{circ}}^i+{\mathbf{F}}_i^T.$$

Full text

I Introduction

A variety of systems can be described as assemblies of particles that exhibit chiral or circular motion 1 ; 2 , such as circularly moving colloids 3 ; 4 ; 5 ; 6 , biological circle swimmers 7 , active spinners 8 ; 9 ; 10 ; 11 ; 12 , circularly driven particles 13 ; 14 ; 15 ; N , and chiral robot swarms 16 . Other systems in which chiral or gyroscopic motion occurs include skyrmions in chiral magnets 17 ; 18 and classical charged particles moving in a magnetic field 19 . Such chiral particle assemblies can exhibit a variety of dynamical phases such as large scale rotations 16 , self-assembly 5 ; 6 ; 8 ; 9 ; 11 , edge currents 5 ; 9 ; 15 , and odd viscosity responses 10 .

Viscosity, fluctuations, and jamming in particle assemblies can be examined at the local level using active rheology, which is based on the response of a probe particle that is driven at either constant force or constant velocity through a fluid or jammed medium 20 ; 21 ; 22 ; 23 ; 24 ; 25 . Active rheology has been applied to the onset of jamming 21 ; 22 ; 26 ; 27 ; 28 ; 29 , where the threshold for probe particle motion increases from zero to a finite value at the jamming transition. It has been used to measure changes in viscosity and diffusive responses 21 ; 30 ; 31 ; 32 ; 33 ; 34 ; 35 as well as velocity-force relations 20 ; 21 ; 27 ; 36 ; 37 ; 38 ; 39 . Active rheology has been applied not only to soft matter systems, but also to the dynamics of individual vortices dragged across pinning landscapes in type-II superconductors 40 ; 41 ; 42 . In systems that are active rather than passive, active rheology shows large changes in the velocity of the probe particle as a function of increasing bath activity when the system transitions from a fluid state to an actively phase separated state 43 . In each case, when the probe particle is driven at constant force, it moves in the direction of drive and exhibits symmetric fluctuations in the transverse direction, with no transverse drift or Hall velocity.

Here we study the active rheology of a probe particle driven through a chiral fluid of circularly swimming disks. We find that for low and intermediate fluid densities, the probe particle exhibits a longitudinal velocity $\langle V_{\rm long}\rangle$ in the direction of drive as well as a finite transverse or Hall velocity $\langle V_{\rm trans}\rangle$, giving rise to a Hall effect with a Hall angle of $\theta_{\rm Hall}=\arctan(\langle V_{\rm trans}\rangle/\langle V_{\rm long}\rangle)$. We examine the evolution of the Hall angle as a function of applied driving force, temperature, and chiral fluid density, and find Hall angles that are as large as $\theta_{\rm Hall}=45^{\circ}$. We also observe non-monotonic behavior of $\theta_{\rm Hall}$ in which the transverse velocity is maximized when a commensuration occurs between the chiral disk rotation frequency and the time interval between consecutive collisions of the probe particle with the chiral disks. In general, $\theta_{\rm Hall}$ decreases with increasing chiral disk density, and it drops to zero at high densities when a jammed state appears. In the dense limit, the probe particle can move only when the driving force is larger than a finite threshold value, and this threshold depends strongly on the chiral disk swimming frequency. At low frequencies, the threshold is nearly zero, while at high frequencies, the threshold increases when the system acts like a solid. We compare the dynamics of the probe particle to driven skyrmions which have recently been shown to exhibit a skyrmion Hall effect that also exhibits nonmonotonic behavior as a function of dc drive, temperature, and skyrmion density 44 ; 45 ; 46 ; 47 ; 48 .

II Simulation and System

We consider a two-dimensional $L\times L$ system with $L=36$ in which we place $N$ non-overlapping disks with a radius $R_{d}=0.5$, where the disks have repulsive harmonic interactions. The density of the system is characterized by the area covered by the disks, $\rho=N\pi R^{2}_{d}/L^{2}$. For monodisperse disks at $T=0$, when $\rho=0.9$ the system forms a triangular solid in which the disks are just touching. The force between disks $i$ and $j$ is given by ${\bf F}^{ij}_{pp}=k(r_{ij}-2R_{d})\Theta(r_{ij}-2R_{d}){\bf\hat{r}}_{ij}$, where $r_{ij}=|{\bf r}_{i}-{\bf r}_{j}|$, ${\bf\hat{r}}_{ij}=({\bf r}_{i}-{\bf r}_{j})/r_{ij}$, and $\Theta$ is the Heaviside step function. The spring stiffness is set to $k=50$, ensuring that the maximum overlap between disks is less than one percent. The densities and parameters we consider here have also been studied in previous works 15 ; 27 ; 43 . The dynamics of disk $i$ is determined by the following overdamped equation of motion:

[TABLE]

We set the damping constant $\eta=1$ and our simulation time step is $\Delta t=0.002$. Here ${\bf F}_{\rm circ}^{i}$ is a driving force that creates a circular motion of the disks of the form ${\bf F}_{\rm circ}^{i}=A(\sin(\omega t){\bf x}+\cos(\omega t){\bf y})$, controlled by varying the drive amplitude $A$. All of the chiral disks move in phase with each other. The thermal force $F^{T}$ is produced by Langevin kicks with the properties $\langle F^{T}_{i}(t)\rangle=0$ and $\langle F^{T}_{i}(t)F^{T}_{j}(t^{\prime})\rangle=2\eta k_{B}T\delta_{ij}\delta(t-t^{\prime})$. Unless otherwise noted, we fix $F^{T}=2.0$, a value large enough to maintain the system in a liquid state up to the solidification density $\rho=0.9$. To create our probe particle, we select a single disk and replace ${\bf F}^{i}_{\rm circ}$ with ${\bf F}_{D}=F_{D}{\bf{\hat{x}}}$. We measure the average velocity response in the longitudinal direction, $\langle V_{\rm long}\rangle=\sum_{i}^{T}{\bf v}_{p}(t_{i})\cdot{\bf\hat{x}}$, as well as in the transverse direction, $\langle V_{\rm trans}\rangle=\sum_{i}^{T}{\bf v}_{p}(t_{i})\cdot{\bf\hat{y}}$, where ${\bf v}_{p}(t_{i})$ is the instantaneous velocity of the probe particle. These quantities are averaged over an interval of $T=5\times 10^{6}$ time steps, which is long enough to ensure that the system has reached a steady dynamical state for the parameters we consider. In the absence of collisions between the probe particle and the chiral disks, we obtain the free flow value $\langle V_{\rm long}\rangle=F_{D}/\eta$.

III Results

In Fig. 1 we show an image of the system highlighting the chiral disk locations and trajectories over a fixed period of time for a system with $\rho=0.181$, $A=2.5$, $\omega=0.006$, and a thermal force of $F^{T}=2.0$. The disks execute circular orbits, and the center of mass of each circular orbit has a diffusive behavior. The red disk is the probe particle, which does not experience a circular drive but instead moves under a force $F_{D}$ applied in the $x$ direction, as indicated by the arrow.

In Fig. 2(a,b) we plot $\langle V_{\rm long}\rangle$ and $\langle V_{\rm trans}\rangle$, respectively, versus $F_{D}$ in a system with $\rho=0.424$, $A=2.5$, and $\omega=0.006$. Here $\langle V_{\rm long}\rangle$ monotonically increases with increasing $F_{D}$ and there is no threshold for motion, while $\langle V_{\rm trans}\rangle$ increases with increasing drive at low $F_{D}$ before reaching a maximum near $F_{D}=1.25$ and then decreasing again. Since both the longitudinal and transverse velocities are finite, the driven particle is moving at an angle with respect to drive direction, similar to the Hall effect found for the motion of a charged particle in a magnetic field. We plot the Hall angle $\theta_{\rm Hall}=\arctan(\langle V_{\rm trans}\rangle/\langle V_{\rm long}\rangle)$ versus $F_{D}$ in Fig. 2(c). The maximum value of $\theta_{\rm Hall}=23^{\circ}$ occurs at $F_{D}=0.75$, a drive smaller than the value of $F_{D}=1.26$ at which the maximum in $\langle V_{\rm trans}\rangle$ appears. For higher drives, $\theta_{\rm Hall}$ gradually deceases, reaching a value close to zero for $F_{D}>4.0$.

In Fig. 3 we illustrate the trajectory of the probe particle over a fixed time interval superimposed on a snapshot of the instantaneous chiral disk locations for the system in Fig. 2 at $F_{D}=1.0$. The probe particle is moving at an angle of approximately $\theta_{\rm Hall}=20^{\circ}$ with respect to the drive; however, there are local trajectory segments in which the Hall angle is larger or smaller than average.

The chiral disks have an intrinsic rotation frequency of $\omega$, and therefore the time required for each chiral disk to complete one orbit is $\tau_{P}=2\pi/\omega$. The average spacing between chiral disks is $a=1/\sqrt{\rho}$. When the probe particle comes into contact with a chiral disk at small $F_{D}$, the chiral disk can complete multiple rotations during the time required for the probe particle to move out of interaction range since $F_{D}\tau_{P}\ll a$. As a result, the average transverse force exerted on the probe particle by the chiral disk is small and $\theta_{\rm Hall}$ is nearly zero. At high $F_{D}$, the probe particle is moving rapidly in the longitudinal direction and spends a very short time interacting with the chiral disks during a collision since $F_{D}\tau_{p}\gg a$, so once again the maximum transverse shift experienced by the probe particle is small and the Hall angle is small. Between these limits, a resonance can occur. When $\omega=0.006$ and $\rho=0.424$, as in Fig. 2, the average spacing between chiral disks is $a=1.35$, and $\tau_{P}=1047\Delta t$, where $\Delta t=0.002$ is the size of a simulation time step. At $F_{D}=0.75$, the probe particle would move a distance $F_{D}\tau_{p}=1.57$ during one chiral rotation period in the absence of collisions with the chiral disks. Collisions reduce this travel distance to a value that is close to $a$, so that on average the probe particle interacts with a given chiral disk for one rotation period. This maximizes the chance that the chiral disk will exert a coordinated, monodirectional transverse force on the probe particle, resulting in a significant transverse displacement. The maximum in $\theta_{\rm Hall}$ thus arises due to a resonance between the chiral rotation time scale and the collision time scale. For higher $\omega$ at the same chiral disk density $\rho$, the peak in $\theta_{\rm Hall}$ shifts to higher values of $F_{D}$. We can compare these results to the behavior of $\theta_{\rm Hall}$ for driven skyrmions 44 ; 46 ; 47 . In the absence of pinning, $\theta_{\rm Hall}$ for the skyrmion system has a constant value determined by the materials properties 44 . When pinning is present, $\theta_{\rm Hall}$ gradually increases from zero at small $F_{D}$, similar to what appears in Fig. 2(b). In the skyrmion case, $\theta_{\rm Hall}$ saturates to the intrinsic value at large $F_{D}$, while for the chiral liquid, $\theta_{\rm Hall}$ decreases as $F_{D}$ increases above the peak value.

In Fig. 4(a) we plot $\langle V_{\rm long}\rangle$ and $\langle V_{\rm trans}\rangle$ versus $F_{D}$ for a non-chiral fluid with the same parameters as in Fig. 2 but with $A=0$. Here, $\langle V_{\rm long}\rangle$ increases monotonically with increasing $F_{D}$, similar to the chiral system; however, $\langle V_{\rm trans}\rangle=0$ for all values of $F_{D}$, indicating that $\theta_{\rm Hall}=0$ and that it is the chiral motion of the bath particles that produces the Hall effect. We find that $\langle V_{\rm long}\rangle$ is slightly lower in the chiral liquid than for the $A=0$ passive disks, as shown in Fig. 4(b) where we compare the $\langle V_{\rm long}\rangle$ versus $F_{D}$ curves for the $A=0$ system from Fig. 4(a) and the $A=2.5$ system from Fig. 2(a). The $A=2.5$ curve is lower for all $F_{D}$, indicating that the chirality of the bath particles increases the longitudinal drag on the probe particle.

In Fig. 5(a) we plot $\langle V_{\rm long}\rangle$ and $\langle V_{\rm trans}\rangle$ versus $\omega$ for a system with $\phi=0.424$, $A=2.5$, and $F_{D}=1.0$. Here there is a dip in $\langle V_{\rm long}\rangle$ near $\omega=0.008$ that coincides with a peak in $\langle V_{\rm trans}\rangle$. The corresponding $\theta_{\rm Hall}$ versus $\omega$ appears in Fig. 5(b), where the Hall angle reaches a maximum value of $\theta_{\rm Hall}=23^{\circ}$ near $\omega=0.004$. At low frequencies, the chiral disks are rotating so slowly that the response is close to that of a passive fluid, while at high frequencies, the chiral orbits diminish in radius and the system again behaves like a passive fluid.

Figure 6 shows $\theta_{\rm Hall}$ versus $\omega$ for the system in Fig. 5 at $F_{D}$ values ranging from $0.125$ to $4.0$. The peak in $\theta_{\rm Hall}$ shifts to higher values of $\omega$ with increasing $F_{D}$ since the chiral particles must rotate faster in order to meet the resonance condition $F_{D}\tau_{P}\sim a$ as $F_{D}$ increases. The maximum value of $\theta_{\rm Hall}$ increases with decreasing $F_{D}$ since the lower longitudinal velocity of the probe particle at the peak in $\theta_{\rm Hall}$ produces a longer collision time and thus a greater transfer of momentum from the chiral disks to the probe particle. The maximum Hall angle we observe at very low $F_{D}$ is close to $\theta_{\rm Hall}=45^{\circ}$.

In Fig. 7(a) we plot the trajectory of the probe particle and the positions of the chiral bath particles for the system in Fig. 6 at $F_{D}=0.25$ and $\omega=0.003$ where $\theta_{\rm Hall}\approx 45^{\circ}$. During some time intervals, the probe particle moves at an angle of nearly $90^{\circ}$ with respect to the driving direction. Figure 7(b) illustrates the same sample at $F_{D}=2.0$ and $\omega=0.012$, where the Hall angle is much smaller, $\theta_{\rm Hall}=6.5^{\circ}$.

In Fig. 8(a) we show $\langle V_{\rm trans}\rangle$ and $\langle V_{\rm long}\rangle$ versus $A$ for a system with $\omega=0.006$, $F_{D}=1.0$ and $\phi=0.424$. Here $\langle V_{\rm long}\rangle$ is large in the $A=0$ passive limit, and it decreases with increasing $A$, passing through a local minimum near $A=7.0$. We find that there is a threshold value $A_{c}=0.5$ below which $\langle V_{\rm trans}\rangle=0$ and there is no transverse response, while a local maximum in $\langle V_{\rm trans}\rangle$ appears at $A=4.0$. We plot the corresponding $\theta_{\rm Hall}$ versus $A$ in Fig. 8(b), where the maximum value of $\theta_{\rm Hall}=27^{\circ}$ occurs near $A=4.0$.

In Fig. 9(a) we plot $\langle V_{\rm long}\rangle$ and $\langle V_{\rm trans}\rangle$ versus $\phi$ for a system with $A=2.5$, $F_{D}=1.0$ and $\omega=0.006$, while in Fig. 9(b) we plot the corresponding $\theta_{\rm Hall}$ versus $\phi$. At the lowest densities, the probe particle undergoes few collisions and moves in the free flow limit with $\langle V_{\rm long}\rangle/F_{D}=1.0$ and $\langle V_{\rm trans}\rangle=0$. As $\phi$ increases, the probe particle velocity gradually decreases, dropping to zero near $\phi=0.86$ which is the effective jamming density for these parameters. The decrease in the mobility of the probe particle with increasing density and the vanishing of the mobility as a jamming or crystallization density is approached resembles what was found in previous studies of active rheology for non-chiral passive disk systems 21 ; 22 ; 26 ; 27 . In those studies, $\langle V_{\rm trans}\rangle=0$ for all values of $\phi$; however, for the chiral disks we find an increase in $\langle V_{\rm trans}\rangle$ with increasing density at low values of $\phi$, with a maximum in $\langle V_{\rm trans}\rangle$ appearing near $\phi=0.35$. This low density increase in the transverse response results from the increasing frequency of collisions between the probe particle and the chiral disks, since it is these collisions that are responsible for the transverse probe particle motion. For $\phi>0.35$, $\langle V_{\rm trans}\rangle$ decreases with increasing density due to a crowding effect, and at jamming $\langle V_{\rm trans}\rangle$ drops to zero. The maximum value of $\theta_{\rm Hall}$ occurs at a higher density of $\phi=0.55$.

We next consider the effect of changing the magnitude of the thermal fluctuations. In Fig. 10(a) we plot $\langle V_{\rm long}\rangle$ and $\langle V_{\rm trans}\rangle$ versus $F^{T}$ for a system with $F_{D}=1.0$, $\omega=0.006$, $A=2.5$, and $\phi=0.424$. At $F^{T}=0$, when the system is in the granular limit, the probe particle leaves a low density wake behind it and $\langle V_{\rm trans}\rangle$ remains finite, indicating that thermal fluctuations are not necessary to produce the Hall response. In Fig. 10(a), $\langle V_{\rm trans}\rangle$ monotonically decreases with increasing $F^{T}$; however, there is a local minimum in $\langle V_{\rm long}\rangle$ near $F^{T}=2.0$. The local minimum roughly coincides with the crossover between low temperatures, where the probe leaves behind a low density wake, and higher temperatures, where the wake rapidly refills with chiral disks. Figure 10(b) shows that the corresponding $\theta_{\rm Hall}$ versus $F^{T}$ has its maximum value at $F^{T}=0$, with a smaller local maximum appearing near $F^{T}=2.0$. Above $F^{T}=2.0$, $\theta_{\rm Hall}$ decreases monotonically with increasing $F^{T}$.

In Fig. 11(a) we illustrate the probe particle motion for the system in Fig. 10(a) with $\phi=0.424$ at $F^{T}=0$, where the probe particle moves at a finite Hall angle and leaves a low density wake in its path. The appearance of an empty region behind the probe particle is similar to what has been observed for active rheology of non-thermal granular materials below the jamming density 22 ; 27 ; 28 , since in these systems there is no energy penalty for the formation of a void. At finite $F^{T}$, the chiral disks diffusively fill in the empty space. In Fig. 11(b) we show the probe particle motion over the same time interval in a denser system with $F^{T}=0$ and $\phi=0.67$. The probe particle does not translate as far due to the decrease in mobility; however, it still leaves behind a low density wake.

In Fig. 10(c) we plot $\langle V_{\rm long}\rangle$ and $\langle V_{\rm trans}\rangle$ versus $F^{T}$ for a high density system with $\phi=0.8482$, $\omega=0.006$, $A=2.5$, and $F_{D}=1.0$, and in Fig. 10(d) we show the corresponding $\theta_{\rm Hall}$ versus $F^{T}$. These parameters fall within a low mobility regime, where the probe particle is not stuck but can only move relatively slowly through the chiral bath. At $F^{T}=0$, $\langle V_{\rm long}\rangle\approx 0.2$ and $\theta_{\rm Hall}=4.5^{\circ}$. As $F^{T}$ increases, both $\langle V_{\rm long}\rangle$ and $\langle V_{\rm trans}\rangle$ decrease, reaching a value that is close to zero near $F^{T}=4.0$. This is a signature of a thermally-induced jamming transition that occurs when the thermal fluctuations increase the effective size of the bath particles and raise the effective density of the system to the jamming density. Such a transition can also be regarded as an example of a freezing by heating phenomenon in which the thermal fluctuations can effectively freeze the disks into a jammed state 49 . If the thermal fluctuations are finite but small, the chiral disks maintain their ordering in the jammed state and the probe particles slowly makes its way through the resulting mostly triangular solid. As $F^{T}$ increases, the fluctuations become strong enough to melt the chiral disk crystal. As a result, liquid behavior reappears and the probe particle mobility rebounds, leading to the increase in both $\langle V_{\rm long}\rangle$ and $\langle V_{\rm trans}\rangle$ for $F^{T}>6.0$. The Hall angle $\theta_{\rm Hall}$ in Fig. 10(d) passes through a local maximum near $F^{T}=2.5$ just before the onset of thermally-induced jamming, and it drops nearly to zero within the jammed state when the probe particle motion becomes extremely slow. For $F^{T}>6.5$, when the jammed state melts and the probe particle mobility increases, $\theta_{\rm Hall}$ increases back to its pre-jammed level. These results indicate that a finite Hall effect can be observed even in non-thermal chiral systems.

Near the jammed state at high chiral disk densities, the behavior of $\theta_{\rm Hall}$ depends strongly on $F_{D}$ and $\omega$, since the probe particle can only move through the jammed chiral disks if the driving force is larger than a depinning threshold $F_{c}$. A monodisperse assembly of passive disks at $T=0$ forms a triangular solid at a density of $\phi=0.9$. For densities close to but below this solidification density, the addition of thermal fluctuations can induce freezing by heating or the formation of grain boundaries and other defects, and the disks exhibit glassy or very slow dynamics for densities in the range $0.83<\phi<0.9$. In our chiral disk system at $F^{T}=2.0$ and $\phi=0.8482$, the probe particle is mobile when $F_{D}=1.0$, but if we reduce $F_{D}$ we find that there is a finite threshold drive $F_{c}$ below which the probe particle is no longer able to move. This is illustrated in Fig. 12(a), where we plot $\langle V_{\rm long}\rangle$ and $\langle V_{\rm trans}\rangle$ versus $F_{D}$ for a system with $\phi=0.8482$, $\omega=0.006$, $F^{T}=2.0$, and $A=2.5$. Here $\langle V_{\rm long}\rangle=\langle V_{\rm trans}\rangle=0$ when $F_{D}<F_{c}$, where the threshold force $F_{c}=0.6$. In the corresponding $\theta_{\rm Hall}$ versus $F_{D}$ curve shown in Fig. 12(b), we find that $\theta_{\rm Hall}=0$ for $F_{D}<0.8$, indicating that within the window $F_{c}<F_{D}<0.8$, the probe particle has a finite longitudinal velocity but exhibits no Hall effect. The Hall angle reaches its maximum value of $\theta_{\rm Hall}\approx 8.5$ near $F_{D}=1.75$, and gradually decreases toward zero for higher drives. Since this system is at a finite temperature of $F^{T}=2.0$, the probe particle is best described as undergoing a creep behavior at $F_{D}=0.8$. During long periods of time, the probe particle is pinned, but there are occasional events in which the probe particle jumps to a new pinned location. Recent studies of driven skyrmions 46 ; 50 showed that the Hall angle is zero in the creep regime and becomes finite at higher drives when the skyrmions transition to continuous flow, similar to what we observe in Fig. 12; however, in the skyrmion case, $\theta_{\rm Hall}$ saturates to the intrinsic value at high drives rather than decreasing back to zero as in Fig. 12.

At high densities, we find that the threshold force $F_{c}$ depends on the frequency at which the chiral disks rotate. In Fig. 13(a) we plot $\langle V_{\rm long}\rangle$ versus $F_{D}$ in a system with $A=2.5$, $F^{T}=2.0$, and $\phi=0.8482$ at $\omega=0.008$, $0.006$, $0.003$, and $0.001$. The threshold for motion is finite at $\omega=0.006$ and $\omega=0.008$, and zero for $\omega=0.003$ and $\omega=0.001$. In Fig. 13(b) we show $F_{c}$ versus $\omega$ for the system from Fig. 13(a). For drives above $F_{c}$, the probe can flow through the sample, but for drives below $F_{c}$, the probe particle is pinned. Here we find that $F_{c}$ is finite only when $\omega>0.003$, and that there is a local maximum in $F_{c}$ near $\omega=0.01$. These results indicate that at higher disk densities, the activity level of the chiral disks can be used to control a transition from jammed to unjammed behavior.

IV Summary

We have numerically examined the motion of a probe particle driven through a chiral liquid composed of circularly moving disks. In the absence of chirality, the probe particle drifts only in the direction of drive so there is no Hall effect; however, when the bath particles are chiral, both the longitudinal and transverse velocities of the probe particle are finite. Since a portion of the probe particle motion is perpendicular to the drive direction, the probe particle exhibits a finite Hall angle similar to what is observed for a charged particle moving in a magnetic field or for driven skyrmion systems. We find that the Hall angle has a non-monotonic dependence on the probe particle driving force and the amplitude and frequency of the chiral disk motion. At low drives, the probe particle can undergo multiple collisions with an individual chiral bath particle, reducing the Hall angle, while at high drives the collisions between the probe and chiral bath particles are very brief, which again reduces the magnitude of the Hall angle. An optimal Hall angle occurs when the time between collisions of the probe particle with consecutive chiral bath particles is roughly equal to the time required for a chiral bath particle to complete a single rotation. We find that the Hall angle can reach values as large as $\theta_{\rm Hall}=45^{\circ}$ and that the Hall effect persists in the zero temperature or granular limit. When the chiral disk activity is fixed, the Hall angle is maximized at an optimal chiral disk density, while the probe particle velocity in both the longitudinal and transverse directions drops to zero when the chiral disks reach the jamming density, which is dependent on the frequency of the chiral motion. At low frequencies, the depinning threshold is zero and the probe particle is able to move under all applied drives, while at higher frequencies there is a finite depinning threshold, and for drives below this threshold, the probe particle is pinned. We compare our results with those obtained for skyrmions moving over random disorder, where drive-dependent Hall angles that increase with increasing drive are observed. In the skyrmion case, the Hall angle saturates to the clean limit at high drives, whereas for the chiral liquid we consider here, the Hall angle decreases to zero at high drives.

Note added– In the course of completing this work, we became aware of the work of Kumar et al. 51 on the motion of spinning probe particles through granular matter, where they report the onset of a Magnus like effect including a lift force.

Acknowledgements.

We gratefully acknowledge the support of the U.S. Department of Energy through the LANL/LDRD program for this work. This work was supported by the US Department of Energy through the Los Alamos National Laboratory. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the U. S. Department of Energy (Contract No. 892333218NCA000001).