Physical Review E 100, 012604 (2019).

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 [3–6], biological circle swimmers [7], active spinners [8–12], circularly driven particles [13–17], and chiral robot swarms [18]. Other systems in which chiral or gyroscopic motion occurs include skyrmions in chiral magnets [19, 20] and classical charged particles moving in a magnetic field [21]. Such chiral particle assemblies can exhibit a variety of dynamical phases such as large scale rotations [18], self-assembly [5, 6, 8, 9, 11], edge currents [5, 9, 15, 22], and odd viscosity responses [10, 23, 24].

Damping, 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 [25–30]. Active rheology has been applied to the onset of jamming [26, 27, 31–34], 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 [26, 35–40] as well as velocity-force relations [25, 26, 32, 41–44]. 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 [45–47]. 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 [48]. 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 ⟨V _long⟩ in the direction of drive as well as a finite transverse or Hall velocity ⟨V _trans⟩, giving rise to a Hall effect with a Hall angle of θ_Hall = arctan(⟨V _trans⟩∕⟨V _long⟩). 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 θ_Hall = 45^∘. We also observe non-monotonic behavior of θ_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, θ_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 [49–53].

II. Simulation and System

We consider a two-dimensional L×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, ϕ = NπR_d²∕L². For monodisperse disks at T = 0, when ϕ = 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 F_pp^ij = k(r_ij - 2R_d)Θ(r_ij - 2R_d) _ij, where r_ij = |r_i - r_j|, _ij = (r_i - r_j)∕r_ij, and Θ 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, 32, 48]. The dynamics of disk i is determined by the following overdamped equation of motion:

(1)

We set the damping constant η = 1 and our simulation time step is Δt = 0.002. Here F_circⁱ is a driving force that creates a circular motion of the disks of the form F_circⁱ = A(sin(ωt) + cos(ωt)ŷ), 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 ⟨F_i^T(t)⟩ = 0 and ⟨F_i^T(t)F_j^T(t^′)⟩ = 2ηk_BTδ_ijδ(t - t^′). 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 ϕ = 0.9. We note that the thermal kicks cause each particle to undergo diffusive behavior at long time scales, which could be appropriate for many types of active colloidal systems. To create our probe particle, we select a single disk and replace F_circⁱ with F_D = F_D. We measure the average velocity response in the longitudinal direction, ⟨V _long⟩ = ∑ _i^T_av_p(t_i) ⋅, as well as in the transverse direction, ⟨V _trans⟩ = ∑_i^T_av_p(t_i) ⋅ŷ, where v_p(t_i) is the instantaneous velocity of the probe particle. These quantities are averaged over an interval of T_a = 5 × 10⁶ 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 ⟨V _long⟩ = F_D∕η.

We focus on two regimes. The first is well below the jamming density at ϕ = 0.424 and F^T = 2.0, where the system forms a liquid state, and the second is in a high density regime at ϕ = 0.8482, close to the jamming limit, where the disks exhibit a finite depinning threshold below which motion does not occur. In the low density regime, we consider a finite chiral activity with A = 2.5 and ω = 0.006. Here the effects of the active rotation are maximized near F_D = 1.0 when the active disks undergo one rotation during the mean interval between collisions with the probe particle. We also study the passive A = 0 case with the same parameters. We examine the effects of varying A, ω, F_D, and F^T, as well as the variation of the density ϕ for fixed activity, and in all cases we show that the transverse response can be maximized at an optimum parameter value.

III. Results

FIG. 1: Instantaneous positions (dark blue circles) and trajectories (light blue lines) of chiral disks during a fixed period of time along with the position (red circle) and driving direction (red arrow) of the probe particle in a sample with ϕ = 0.181, A = 2.5, ω = 0.006, F^T = 2.0, and F _D = 2.0. The chiral disks undergo a combination of diffusion and circular motion.

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 ϕ = 0.181, A = 2.5, ω = 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.

FIG. 2: Local probe response in a system with ϕ = 0.424, A = 2.5 and ω = 0.006. (a) The longitudinal velocity ⟨V _long⟩ vs F_D. (b) The transverse velocity ⟨V _trans⟩ vs F_D. (c) The Hall angle θ_Hall = tan ^-1(⟨V _trans⟩∕⟨V _long⟩) vs F_D.

In Fig. 2(a,b) we plot ⟨V _long⟩ and ⟨V _trans⟩, respectively, versus F_D in a system with ϕ = 0.424, A = 2.5, and ω = 0.006. Here ⟨V _long⟩ monotonically increases with increasing F_D and there is no threshold for motion, while ⟨V _trans⟩ 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 θ_Hall = tan^-1(⟨V _trans⟩∕⟨V _long⟩) versus F_D in Fig. 2(c). The maximum value of θ_Hall = 23^∘ occurs at F_D = 0.75, a drive smaller than the value of F_D = 1.26 at which the maximum in ⟨V _trans⟩ appears. For higher drives, θ_Hall gradually deceases, reaching a value close to zero for F_D > 4.0.

FIG. 3: The longitudinal motility M_long = ⟨V _long⟩∕F_D vs F_D for the active system in Fig. 1 (dark blue triangles) and for a passive system with A = 0 (light blue circles). Also plotted is the transverse mobility M_trans = ⟨V _trans⟩∕F_D vs F_D for the same active (dark red triangles) and passive (light red circles) systems. (b) The effective damping η_eff obtained from the net velocity ⟨V ⟩ = (⟨V _trans⟩² + ⟨V _long⟩²)^1∕2 for the active (dark brown triangles) and passive (light brown circles) systems. The damping is enhanced in the active system, particularly at low drives. The black dashed line indicates the damping η = 1 experienced by an isolated free particle.

We measure the longitudinal mobility M_long = ⟨V _long⟩∕F_D and transverse mobility M_trans = ⟨V _trans⟩∕F_D for the system in Fig. 2 and show the resulting curves in Fig. 3(a). Starting from a small value, M_long increases with increasing F_D until it approaches the free-flow limit of M_long = 1 at high drives. In contrast, M_trans increases to a maximum value of M_trans ≈ 0.2 near F_D = 1.0 and then decreases to M_trans = 0 at high drives. For comparison, in Fig. 3(a) we also plot the behavior of M_long and M_trans in the A = 0 or inactive limit, where M_trans = 0 for all values of F_D. The value of M_long is always higher in the inactive system than in the sample with finite chiral motion, indicating that the chiral motion increases the effective damping or viscosity experienced by the moving probe particle. A current topic in many chiral active matter systems is the question of odd viscosity response [22–24], but it is not clear exactly what the signature of odd viscosity would be in the local driven probe system. In Fig. 3(b) we plot the effective damping constant η_eff = 1∕⟨V ⟩ versus F_D for the active and passive systems shown in Fig. 3(a). Here the net velocity is given by ⟨V ⟩ = (⟨V _trans⟩² + ⟨V _long⟩²)^1∕2. The effective damping for A = 0 is largest at low F_D and decreases monotonically with increasing drive, gradually approaching the free-particle limit of η = 1 at high drives. When we introduce finite activity, we find a large increase in η_eff at low drives F_D < 1. This also means that the viscosity of the active system is larger, indicating that the active rotation increases the net damping in the system.

FIG. 4: Instantaneous positions of chiral disks (dark blue circles) and probe particle (red circle) along with the probe particle trajectory (red line) over a period of time for the system in Fig. 2 with ϕ = 0.424, A = 2.5, and ω = 0.006 at F_D = 1.0, where the probe particle moves at an average Hall angle of θ_Hall = 20^∘.

In Fig. 4 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 θ_Hall = 20^∘ 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 ω, and therefore the time required for each chiral disk to complete one orbit is τ_P = 2π∕ω. The average spacing between chiral disks is a = 1∕. 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_Dτ_P ≪ a. As a result, the average transverse force exerted on the probe particle by the chiral disk is small and θ_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_Dτ_P ≫ 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 ω = 0.006 and ϕ = 0.424, as in Fig. 2, the average spacing between chiral disks is a = 1.35, and τ_P = 1047Δt, where Δt = 0.002 is the size of a simulation time step. At F_D = 0.75, the probe particle would move a distance F_Dτ_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 θ_Hall thus arises due to a resonance between the chiral rotation time scale and the collision time scale. For higher ω at the same chiral disk density ϕ, the peak in θ_Hall shifts to higher values of F_D. We can compare these results to the behavior of θ_Hall for driven skyrmions [49, 51, 52]. In the absence of pinning, θ_Hall for the skyrmion system has a constant value determined by the materials properties [49]. When pinning is present, θ_Hall gradually increases from zero at small F_D, similar to what appears in Fig. 2(b). In the skyrmion case, θ_Hall saturates to the intrinsic value at large F_D, while for the chiral liquid, θ_Hall decreases as F_D increases above the peak value.

FIG. 5: (a) ⟨V _long⟩ (dark blue circles) and ⟨V _trans⟩ (red squares) vs F_D for a non-chiral fluid with the same parameters as in Fig. 2 except with A = 0. Here ⟨V _trans⟩ = 0 for all values of F_D. (b) ⟨V _long⟩ (dark blue circles) vs F_D for the system in panel (a) at A = 0.0 and ⟨V _long⟩ (light blue circles) vs F_D for the chiral system in Fig. 2 with A = 2.5, showing that the damping of the longitudinal motion is larger in the chiral fluid.

In Fig. 5(a) we plot ⟨V _long⟩ and ⟨V _trans⟩ versus F_D for a non-chiral fluid with the same parameters as in Fig. 2 but with A = 0. Here, ⟨V _long⟩ increases monotonically with increasing F_D, similar to the chiral system; however, ⟨V _trans⟩ = 0 for all values of F_D, indicating that θ_Hall = 0 and that it is the chiral motion of the bath particles that produces the Hall effect. We find that ⟨V _long⟩ is slightly lower in the chiral liquid than for the A = 0 passive disks, as shown in Fig. 5(b) where we compare the ⟨V _long⟩ versus F_D curves for the A = 0 system from Fig. 5(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.

FIG. 6: (a) ⟨V _trans⟩ (red squares) and ⟨V _long⟩ (blue circles) vs ω for a system with F_D = 1.0, A = 2.5 and ϕ = 0.424. A minimum in ⟨V _long⟩ coincides with a maximum in ⟨V _trans⟩ near ω = 0.008. (b) The corresponding θ_Hall vs ω showing a maximum Hall angle of θ_Hall = 23^∘ near ω = 0.004.

In Fig. 6(a) we plot ⟨V _long⟩ and ⟨V _trans⟩ versus ω for a system with ϕ = 0.424, A = 2.5, and F_D = 1.0. Here there is a dip in ⟨V _long⟩ near ω = 0.008 that coincides with a peak in ⟨V _trans⟩. The corresponding θ_Hall versus ω appears in Fig. 6(b), where the Hall angle reaches a maximum value of θ_Hall = 23^∘ near ω = 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.

FIG. 7: θ_Hall vs ω for the system in Fig. 6 with A = 2.5 and ϕ = 0.424 for F_D = 0.125 (violet circles), 0.25 (dark blue squares), 0.5 (light blue diamonds), 0.75 (teal up triangles), 1.0 (green left triangles), 2.0 (orange down triangles) and 4.0 (red right triangles). Here the maximum in θ_Hall shifts to higher ω with increasing F_D while the maximum possible Hall angle decreases.

Figure 7 shows θ_Hall versus ω for the system in Fig. 6 at F_D values ranging from 0.125 to 4.0. The peak in θ_Hall shifts to higher values of ω with increasing F_D since the chiral particles must rotate faster in order to meet the resonance condition F_Dτ_P ~ a as F_D increases. The maximum value of θ_Hall increases with decreasing F_D since the lower longitudinal velocity of the probe particle at the peak in θ_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 θ_Hall = 45^∘.

FIG. 8: (a) ω₀, the frequency at which the maximum Hall angle appears, vs F_D for the system in Fig. 7 with A = 2.5 and ϕ = 0.424. There is a roughly linear increase in ω₀ with increasing F_D. (b) θ_Hall^max, the value of the Hall angle at ω ₀, vs F_D for the same system, showing a roughly linear decrease with F_D. (c) θ_Hall∕θ_Hall^max vs ω∕ω ₀, showing a collapse of the curves in Fig. 7 based on the fits in panels (a) and (b). Here, F_D = 0.125 (violet circles), 0.25 (dark blue squares), 0.5 (light blue diamonds), 0.75 (teal up triangles), 1.0 (green left triangles), 2.0 (orange down triangles), and 4.0 (red right triangles). The thick dashed line is a fit to θ_Hall∕θ_Hall^max ∝ 1∕ω^* for ω^* > 0, where ω^*≡ ω∕ω ₀ - 1.

In Fig. 8(a) we plot the frequency ω₀ at which θ_Hall reaches its maximum value versus F_D for the system in Fig. 7. We find that ω₀ increases linearly with increasing F_D since it appears at a frequency for which a resonance occurs between the time required for an active disk to complete a revolution and the time that separates consecutive collisions between the probe particle and the active disks. The collision time is proportional to the probe velocity, which varies linearly with F_D, and therefore the resonant frequency ω₀ also varies linearly with F_D. We note that at lower F_D, where the probe velocity dependence on F_D becomes nonlinear, this behavior breaks down. In Fig. 8(b) we plot θ_Hall^max, the maximum value of the Hall angle, versus F_D. Here θ_Hall^max decreases roughly linearly with increasing F_D, with some deviation from linearity at low values of F_D. Using the linear fits of ω₀ and θ_Hall^max, we can collapse the curves from Fig. 7, as shown in the plot of θ_Hall∕θ_Hall^max versus ω∕ω₀ in Fig. 8(c). The dashed line in Fig. 8(c) is a fit to the form θ_Hall∕θ_Hall^max ∝ 1∕ω^* for ω^* > 0, where ω^*≡ ω∕ω₀ - 1, showing that θ_Hall decays as an inverse power law above the resonant frequency.

FIG. 9: Instantaneous positions of chiral disks (dark blue circles) and probe particle (red circle) along with the probe particle trajectory (red line) over a period of time for the system in Fig. 7 with A = 2.5 and ϕ = 0.424. (a) F_D = 0.25 and ω = 0.003, where θ_Hall ≈ 45^∘. (b) F _D = 2.0 and ω = 0.012, where θ_Hall = 6.5^∘.

In Fig. 9(a) we plot the trajectory of the probe particle and the positions of the chiral bath particles for the system in Fig. 7 at F_D = 0.25 and ω = 0.003, where θ_Hall ≈ 45^∘. During some time intervals, the probe particle moves at an angle of nearly 90^∘ with respect to the driving direction. Figure 9(b) illustrates the same sample at F_D = 2.0 and ω = 0.012, where the Hall angle is much smaller, θ_Hall = 6.5^∘.

FIG. 10: (a) ⟨V _long⟩ (blue circles) and ⟨V _trans⟩ (red squares) vs A for a system with ω = 0.006, F_D = 1.0, and ϕ = 0.424. (b) The corresponding θ_Hall vs A. There is a threshold value of A_c = 0.5 below which θ_Hall = 0 and the Hall response is absent.

In Fig. 10(a) we show ⟨V _trans⟩ and ⟨V _long⟩ versus A for a system with ω = 0.006, F_D = 1.0 and ϕ = 0.424. Here ⟨V _long⟩ 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 ⟨V _trans⟩ = 0 and there is no transverse response, while a local maximum in ⟨V _trans⟩ appears at A = 4.0. We plot the corresponding θ_Hall versus A in Fig. 10(b), where the maximum value of θ_Hall = 27^∘ occurs near A = 4.0.

FIG. 11: (a) ⟨V _long⟩ (blue circles) and ⟨V _trans⟩ (red squares) vs ϕ for a system with ω = 0.006, F_D = 1.0 and A = 2.5. (b) The corresponding θ_Hall vs ϕ. A jamming transition occurs near ϕ = 0.86.

In Fig. 11(a) we plot ⟨V _long⟩ and ⟨V _trans⟩ versus ϕ for a system with A = 2.5, F_D = 1.0 and ω = 0.006, while in Fig. 11(b) we plot the corresponding θ_Hall versus ϕ. At the lowest densities, the probe particle undergoes few collisions and moves in the free flow limit with ⟨V _long⟩∕F_D = 1.0 and ⟨V _trans⟩ = 0. As ϕ increases, the probe particle velocity gradually decreases, dropping to zero near ϕ = 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 [26, 27, 31, 32]. In those studies, ⟨V _trans⟩ = 0 for all values of ϕ; however, for the chiral disks we find an increase in ⟨V _trans⟩ with increasing density at low values of ϕ, with a maximum in ⟨V _trans⟩ appearing near ϕ = 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 ϕ > 0.35, ⟨V _trans⟩ decreases with increasing density due to a crowding effect, and at jamming ⟨V _trans⟩ drops to zero. The maximum value of θ_Hall occurs at a higher density of ϕ = 0.55.

We note that in principle it would be possible to perform a collapse of ⟨V _trans⟩, ⟨V _long⟩, and θ_Hall for varied values of ω, F_D and A, similar to what is shown in Fig. 8. The number of independent variables can be reduced slightly since the behavior as a function of A should be proportional to the behavior as a function of 1∕ω_p, where ω_p ∝ 1∕(F_D + F_D^c) above the resonance and ω_p ∝ 1∕(F_D - F_D^c) below the resonance. Here F_D^c is the value of the dc drive at which the resonance occurs. Although we find that the drift velocity generally increases with F_D, we do not observe unbounded acceleration of the probe particle of the type that can occur in a Fermi acceleration process. This is expected since the inclusion of even a small amount of dissipation can destroy the Fermi acceleration mechanism [54].

FIG. 12: (a) ⟨V _long⟩ (blue circles) and ⟨V _trans⟩ (red squares) vs F^T for a system with ω = 0.006, F _D = 1.0, A = 2.5, and ϕ = 0.424. (b) The corresponding θ_Hall vs F_T. (c) ⟨V _long⟩ (blue circles) and ⟨V _trans⟩ (red squares) vs F^T for a system with ω = 0.006, F_D = 1.0, A = 2.5, and ϕ = 0.8482. (d) The corresponding θ_Hall vs F_T. Here we find a freezing by heating phenomenon in the interval 4.0 < F^T < 7.0.

FIG. 13: Instantaneous positions of chiral disks (dark blue circles) and probe particle (red circle) for the system in Fig. 12(a) with ω = 0.006, F_D = 1.0, A = 2.5, and ϕ = 0.424 at F^T = 0, where the probe particle has a finite Hall angle but leaves a low density depletion zone or wake behind as it moves. (b) The same system at a higher density of ϕ = 0.67. Here the probe particle mobility is reduced but the Hall angle remains finite and the depletion zone persists.

We next consider the effect of changing the magnitude of the thermal fluctuations. In Fig. 12(a) we plot ⟨V _long⟩ and ⟨V _trans⟩ versus F^T for a system with F_D = 1.0, ω = 0.006, A = 2.5, and ϕ = 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 ⟨V _trans⟩ remains finite, indicating that thermal fluctuations are not necessary to produce the Hall response. In Fig. 12(a), ⟨V _trans⟩ monotonically decreases with increasing F^T; however, there is a local minimum in ⟨V _long⟩ 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 12(b) shows that the corresponding θ_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, θ_Hall decreases monotonically with increasing F^T.

In Fig. 13(a) we illustrate the probe particle motion for the system in Fig. 12(a) with ϕ = 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 [27, 32, 33], 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. 13(b) we show the probe particle motion over the same time interval in a denser system with F^T = 0 and ϕ = 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. 12(c) we plot ⟨V _long⟩ and ⟨V _trans⟩ versus F^T for a high density system with ϕ = 0.8482, ω = 0.006, A = 2.5, and F_D = 1.0, and in Fig. 12(d) we show the corresponding θ_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, ⟨V _long⟩≈ 0.2 and θ_Hall = 4.5^∘. As F^T increases, both ⟨V _long⟩ and ⟨V _trans⟩ 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 [55]. If the thermal fluctuations are finite but small, the chiral disks maintain their ordering in the jammed state and the probe particle 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 ⟨V _long⟩ and ⟨V _trans⟩ for F^T > 6.0. The Hall angle θ_Hall in Fig. 12(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, θ_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.

FIG. 14: (a) ⟨V _long⟩ (blue circles) and ⟨V _trans⟩ (red squares) vs F_D for a system with ϕ = 0.848, A = 2.5, and F^T = 2.0. There is a finite depinning threshold near F_D = 0.6. (b) The corresponding θ_Hall vs F_D passes through a maximum at F_D = 1.75.

Near the jammed state at high chiral disk densities, the behavior of θ_Hall depends strongly on F_D and ω, 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 ϕ = 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 < ϕ < 0.9. In our chiral disk system at F^T = 2.0 and ϕ = 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. 14(a), where we plot ⟨V _long⟩ and ⟨V _trans⟩ versus F_D for a system with ϕ = 0.8482, ω = 0.006, F^T = 2.0, and A = 2.5. Here ⟨V _long⟩ = ⟨V _trans⟩ = 0 when F_D < F_c, where the threshold force F_c = 0.6. In the corresponding θ_Hall versus F_D curve shown in Fig. 14(b), we find that θ_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 θ_Hall ≈ 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 [51, 56] 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. 14; however, in the skyrmion case, θ_Hall saturates to the intrinsic value at high drives rather than decreasing back to zero as in Fig. 14.

FIG. 15: (a) ⟨V _long⟩ vs F_D for a system with ϕ = 0.8482, A = 2.5, and F^T = 2.0 at ω = 0.008 (red triangles), 0.006 (green squares), 0.003 (light blue circles), and 0.001 (dark blue diamonds). (b) The depinning force F_c vs ω for the same system highlighting regions in which the probe particle is moving (pink) or pinned (yellow).

At high densities, we find that the threshold force F_c depends on the frequency at which the chiral disks rotate. In Fig. 15(a) we plot ⟨V _long⟩ versus F_D in a system with A = 2.5, F^T = 2.0, and ϕ = 0.8482 at ω = 0.008, 0.006, 0.003, and 0.001. The threshold for motion is finite at ω = 0.006 and ω = 0.008, and zero for ω = 0.003 and ω = 0.001. In Fig. 15(b) we show F_c versus ω for the system from Fig. 15(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 ω > 0.003, and that there is a local maximum in F_c near ω = 0.01. We note that the appearance of a finite depinning threshold for a probe particle has been observed experimentally for systems in a high density or glassy regime [26, 57]. For the passive A = 0 limit in our system, we are always below the non-active jamming density of 0.9, so the inclusion of activity can be viewed as effectively increasing the density of the particles. An open question is what effect the chiral activity would have on a system that is above the passive jamming density. For example, addition of activity could increase or decrease the critical depinning force. Our 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 θ_Hall = 45^∘ 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. Our results could be tested by driving probe particles through active chiral colloidal spinners, chiral granular matter, or even chiral robot swarms. Additionally, these results could be relevant to other systems that mimic chiral active baths, such as driving a single skyrmion through an array of other skyrmions or driving a single particle through an array of optical or fluid vortices.

Note added– In the course of completing this work, we became aware of the work of Kumar et al. [58] on the motion of spinning probe particles through granular matter, where they report the onset of a Magnus like effect including a lift force.

Acknowledgments

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).