Home 3d Printing Emergent stereoselective interactions and self-recognition in polar chiral active ellipsoids

# Emergent stereoselective interactions and self-recognition in polar chiral active ellipsoids

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## INTRODUCTION

When chirality is present as a static feature of building blocks, interactions between and reactions involving them are often stereoselective (1). For molecular building blocks, chirality also leads to higher-order effects such as self-recognition, sorting, and discrimination (2). There are, however, many instances where chirality is also manifest in the dynamics. Examples in the realm of classical physics where particle trajectories have a chirality/handedness associated with them include assemblies of spinning colloidal magnets (3, 4) and granules (5), spinning (6) and circularly swimming bacteria (7) in the vicinity of walls, spiral (8) and helical swimmers (9), and synthetic chiral active particles (1013). In one of these systems—magnetic disks set spinning by a rotating external field—where chirality was present in both the particle shape and dynamics, emergent stereoselective interactions between particles was observed to aid in their dynamic self-assembly (5). A rather fundamental question then is whether such interactions, and maybe even self-recognition, can emerge in a system where chirality is only a feature of the dynamics. Systems like the one studied in (5) are immediately ruled out, since all particles have the same handedness in their dynamics, which is set by the external rotating field, and when the particle shape is now made achiral, stereoselective interactions cannot even arise. These limitations, however, do not apply to active matter systems (14), since, here, the direction of motion, and also the handedness if activity is chiral (15), is set by the particle. In active matter ensembles sans attractive interparticle interactions, even achiral activity can result in notable collective behavior (14) and also lead to condensed phases (16, 17). Even while chiral activity brings further richness to the dynamics (13, 18, 19), with numerical studies also anticipating chiral sorting (20), whether such an activity can mediate selectivity and recognition in these systems remains unanswered.

Addressing this question in experiments, even in two dimensions (2D), requires a system that provides some degree of control over chiral activity, and realizing this is not without its challenges. In 2D, the relevant parameters are the radius of the circular trajectory, R, and the angular velocity, ω, of the particle (15). Tuning either or both is difficult in living chiral active matter. While synthetic active spinners are easy to fabricate (10, 12, 13), being subject to only an active torque, R = 0, this constrains the parameter space of chiral activity. To realize the more generic circular trajectories, where both active torques and forces act, existing strategies for both wet and dry active matter rely on particle shapes that are chiral (10, 11). Thus, within these approaches, disentangling emergent effects associated with chiral activity from those arising due to the chiral particle shape is virtually impossible.

## RESULTS

### Realizing tunable chiral active motion

Here, we achieved tunable chiral active motion while keeping the particle shape and size fixed and achiral by capitalizing on a hitherto unexploited feature of a canonical 2D active matter system—assemblies of millimeter-sized grains rendered active through vertical agitation. Previous experiments found that vibrated granules of simple shapes, like disks or rods, but with an asymmetry in mass, m, and/or friction coefficient, μ, were polar active along the direction set by the asymmetry (2123). While, in these studies, when the asymmetry in both were present they were coincident, in our granules—3D-printed plastic ellipsoids—over and above an asymmetry in μ and m along the major axis, we also had an asymmetry in m along the minor axis. We anticipated vertical agitation to generate both an active force and torque on these particles and converged on this specific design after systematically excluding out other designs based on the nature of active dynamics observed (see fig. S4).

Our Janus ellipsoids had a major axis, α = 6 mm, and minor axes, β = 3 mm and γ = 2.1 mm. We exploited a feature unique to the 3D print process to introduce a fore-aft asymmetry in μ, and this can be seen as a difference in the surface finish between the lower and upper half of the ellipsoid in Fig. 1 (A to F). A left-right asymmetry in m, with respect to the particle major axis, was achieved by making one portion of the ellipsoid hollow during print (shown by dashed red lines in Fig. 1, A to F). The printed particles were placed on a horizontally mounted flower-shaped arena and confined from above with a transparent glass plate, which also enabled imaging of dynamics. The entire assembly was coupled to a permanent magnet shaker through a stiff air bearing, which ensured that the imposed sinusoidal driving from the shaker is channeled only into vertical oscillations of the plate (see Materials and Methods) (24). The drive frequency f = 37 Hz and amplitude a = 1 mm were kept constant in all experiments, unless specified, and the nondimensional acceleration

$Γ=4π2f2ag=5.5$

, where g is the acceleration due to gravity. We quantified the dynamics of individual ellipsoids by working at a very low area fraction ϕ ≈ 0.1% (N ≈ 8 to 10 particles on the plate), where ϕ is the total projected area of all particles on the plate divided by the surface area of the plate. The energy gained by the particles through frequent collisions with the vibrating plate resulted in polar chiral active motion—noisy circular trajectories with a more or less well-defined R and ω (Fig. 1A, right, and movie S1). Since no special care was taken while placing particles on the plate, the granular assemblies were racemic mixtures and had nearly equal numbers of particles with dextrogyre [(+)] and levogyre [(−)] trajectories. Furthermore, as the gap between plates Δ = 2.4 mm is such that γ < Δ < β, particles cannot flip once they are confined, and the handedness of the orbits remained unchanged.

Now, by changing only the volume of the hollowed-out portion of the ellipsoid during 3D printing (red dashed lines in Fig. 1, A to F), we tuned the extent of left-right mass asymmetry, ΔmLR, and hence the active torque (see fig. S5). We tracked the dynamics of many individual ellipsoids for a given ΔmLR and determined the probability distribution of angular displacements P(Δθ) between successive time instances (Δt = 0.03 s). P(Δθ) was Gaussian for all the mass asymmetries studied, and its peak shifted to smaller Δθ with ΔmLR (Fig. 1G). As anticipated, decreasing ΔmLR resulted in a smaller active torque and thus a decrease in ω and an increase in R (right panels in Fig. 1, A to F). Ellipsoids with different ΔmLR values are labeled C1 to C6, where a larger subscript indicates a larger R (see Table 1).

Table 1 Details of various 3D-printed polar chiral active ellipsoids.

### Emergent stereoselective interactions in polar chiral active ellipsoids

Having gained the ability to tune chiral activity, we investigated dynamics of the granular assemblies at higher ϕ. Even at a ϕ as low as 1%, we observed dimer formation solely due to chiral activity. We observed qualitatively similar dynamics for C1 and C2 ellipsoids. In racemic mixtures of these particles, we observed two distinct types of active dimers, which we classified as “movers” and “spinners” based on their dynamics. This difference in behavior was entirely a consequence of their composition. The mover was made of a (+) and a (−) monomer with their polar axis pointing in nearly the same direction, and the adduct largely behaved like an achiral polar active particle (movie S2). Figure 2A shows the representative experimental trajectory of a mover made of C2 monomers. The translational mean squared displacement (MSD) of the center of mass (COM) of the adduct was superdiffusive at small lag times t and then crossed over to diffusive dynamics at larger t (solid black symbols in Fig. 2C) (25). However, the orientational MSD of movers was superdiffusive at longer times, indicating that the movers experienced some residual torque (fig. S6). In contrast, a spinner was made of monomers that were both (+) or both (−) but whose polar axis pointed in opposite directions (see Fig. 2B and movie S3 for a spinner made of C2 monomers). The orientational MSD of the spinners showed a crossover from superdiffusive to ballistic behavior with t (red circles in Fig. 2C). Because of the almost complete cancellation of the active forces, the translational dynamics of the spinner was purely diffusive with a diffusion constant that was nearly two orders of magnitude smaller than that of the movers (fig. S7).

Such mover and spinner states are theoretically predicted for two hydrodynamically interacting active rotors (26). However, it is not known whether, in a racemic mixture of these rotors, one of these bound states is more abundant, which would then indicate active stereoselectivity (2). Hydrodynamic interactions, nevertheless, are not relevant for dry active matter. To determine whether, in our system, interactions were stereoselective, we calculated the fraction of dimers that existed as movers and spinners. In racemic mixtures of C1 and C2 ellipsoids, the mover fraction is nearly three times the spinner fraction—a clear sign of active stereoselectivity (Fig. 2D). For larger R (C3 to C6 ellipsoids), we exclusively observed only movers, although its lifetime decreased with R, since the opposing active torques keeping the monomers together also decreased (movie S4 and fig. S8). The absence of spinners at large R is also understandable. For a spinner to be stable, monomers should not slip past each other, and this is possible only when R is comparable to the radius of revolution of the monomer about the spinners’ COM, which is nothing but

$β2$

. R<

$β2$

for C1 and C2 ellipsoids and R >

$β2$

for C3 to C6 ellipsoids (Table 1).

To understand why movers are more favored over spinners, we investigated the physical mechanisms of dimer formation by closely observing the path traced by the monomers moments before they formed the adduct. Snapshots in Fig. 2 (E to G), show the reaction pathways for the mover and spinner formation modes, respectively. Two reaction mechanisms typically mediate a mover bond; the first one is via “collision followed by realignment” in which the formation of the mover bond is aided by multiple collisions and realignments of the constituent monomers (Fig. 2E). The second mode, which also occurred frequently, is via the disintegration of a trimer into a mover and monomer (Fig. 2F). However, a spinner is formed by a single mechanism, i.e., “two-particle collision.” As opposed to mover formation, realignments are rare, and the spinner was created only for those collision events where the particles were more or less aligned when coming into contact (Fig. 2G). Clearly, the ease with which movers could form over spinners resulted in the observed stereoselective interactions.

### Self-recognition in polar chiral active ellipsoids

We now considered the possibility of this active stereoselective interaction leading to chiral self-recognition (2); i.e., in assemblies made of ellipsoids of very different chiral activities, there was a preference in the way particles paired up. We focused on mixtures of C2 and C4 ellipsoids, as their dynamics were representative of the limiting cases R<

$β2$

and >

$β2$

, respectively. Movie S5 shows the dynamics of a 1 : 1 mixture of individually racemic C2 and C4 ellipsoids at ϕ = 10%. This experiment provided the first cue that there might be some form of self-recognition between particles—we observed C2C2 spinners and never any C4C4 spinners, as expected, but we also never saw any C2C4 spinners. We, however, observed C2C2, C4C4, and also C2C4 movers. Measuring their average lifetimes to determine whether there was self-recognition was difficult at high ϕ owing to their interactions with other particles, and at low ϕ, we could not obtain enough statistics.

To overcome this difficulty, before confining the particles and setting the plate in motion, we manually created one each of the three mover configurations—C2C2, C4C4, and C2C4—by placing the appropriate (+) and (−) monomers side by side (inset to Fig. 3A). This initial configuration helped substantially promote dimerization under vertical agitation (movie S6). Trajectories representative of each of the three movers is shown in Fig. 3B. By repeating this experiment a few hundred times, we measured the probability distribution of mover lifetimes, P(τ), for each of the three movers. P(τ) was an exponential for all dimer types, and we defined the average lifetime as

$P(τ=τmov)=1e$

. These measurements provided the first evidence for chiral self-recognition in active matter: Movers made of like particles were longer-lived in comparison to those made of unlike ones (Fig. 3A). In C2C4 movers, besides the radii, the angular velocities of the constituent monomers are also different (Table 1). The monomers thus slide past each other, and the lifetime of these movers is smaller than even C4C4 ones.

### Relaxation dynamics depends on the net chirality of active liquids

Having uncovered the active matter analogs of stereoselectivity, we next explored the consequences of these interactions in the dense limit. Unlike in assemblies of C3 to C6 ellipsoids, where we never observed spinner states, in liquids of C1 and C2 ellipsoids, both movers and spinners exist. Furthermore, in liquids of C1 and C2 ellipsoids, even at a fixed ϕ, the net chirality, χ, decides whether movers can even form; in a racemic mixture (χ = 0), both mover and spinner states exist, whereas in an enantiopure liquid (χ = 1), only spinners exist. Here, χ = ∣ N+N ∣ /N+ + N, where N+ and N are the fraction of the (+) and (−) monomers, respectively. Since movers and spinners have very distinct dynamics, we naïvely expect that at a fixed ϕ, varying χ alters the nature of relaxation dynamics.

To begin with, we looked for quantifiable differences in the relaxation dynamics of liquids of C2 ellipsoids across three different net chiralities: χ = 1 [100% (+) monomers], χ = 0.5 [75% (+) monomers and 25% (−) monomers], and χ = 0 [50% (+) monomers and 50% (−) monomers]. We varied ϕ from 0.1 to 0.84 while ensuring that, across χ, the ϕ′s were nearly identical to enable comparison. For all χ, we observed the formation of dynamic aggregates for ϕ ⪆ 0.2 (figs. S9 and S10). We, however, did not observe any large-scale phase separation consistent with recent simulation findings on active rods (27, 28). Also, over this range of χ and ϕ values studied, we did not observe segregation of enantiomers (13), microflocks with phase-locked dynamics (18, 29), and activity-induced synchronization (20). There was no notable difference in the static or dynamical properties of these clusters with χ, and we do not dwell on it here. Next, we quantified the structural relaxation time for the orientational degrees of freedom (DOF),

$ταR$

, and the translational DOF,

$ταT$

, through their respective time correlators (fig. S11). Like in passive liquids of ellipsoids (30), even for our chiral active liquids, at fixed χ and with ϕ, orientational slowing down is more rapid than the translational one—

$ταR$

grows by more than two orders of magnitude while

$ταT$

grows by one order of magnitude (Fig. 4, A and B, and fig. S12). The isochrones (dashed white lines in Fig. 4, A and B), however, are not parallel to the ordinate but appear curved, suggesting that, at constant ϕ, changing χ alters relaxation dynamics. While interesting, this inference needs to be interpreted with caution due to the limited χ values explored hitherto. This is mainly because the enantiopure (χ = 1) and enantiomeric excess (χ = 0.5) liquids had to be made by manually placing each monomer of the correct handedness, almost 8000 particles for ϕ = 0.84, on the plate.

To strengthen these findings, for ϕ = 0.72, we carried out experiments for six values of χ. The effect of changing χ on relaxation dynamics is now clear—both

$ταR$

(Fig. 4C) and

$ταT$

(Fig. 4D) increase almost linearly with χ, albeit the change in the latter is smaller than in the former. Although the overall growth in

$ταR$

and

$ταT$

is by ≈20 and ≈10%, respectively, we emphasize that this stems purely from changing χ alone and is indeed unexpected. Since we observed no apparent change in the structure of these liquids, we sought to determine whether the observed slowing down with χ was a consequence of growing dynamical length scales. This line of thinking was motivated by studies on dense supercooled liquids where an increase in the structural relaxation times with density, or on lowering temperature, is accompanied by a growing cooperativity length scale, associated with the size of dynamical heterogeneities, but without any apparent change in static structure (31, 32). Borrowing from these studies, we determined whether dynamics even in our chiral active liquids was spatiotemporally heterogeneous. At ϕ = 0.72 and for all χ, we quantified the top 20% of the least mobile particles over a time interval 5t*, where t* is the cage rearrangement time (see fig. S13). We found that density relaxation was dynamically heterogeneous and that the least mobile particles were spatially clustered (Fig. 4E). Two least mobile particles were said to belong to the same cluster if one ellipsoid, when expanded 1.4 times the major axis, encompasses the other’s center. We then divided the entire experimental data into six statistically independent realizations, and Fig. 4F shows the probability distribution of cluster sizes P(Nc) for one such realization. From P(Nc), we computed the average size of cooperatively rearranging regions

$⟨NcT⟩=ΣNc2P(Nc)ΣNcP(Nc)$

. 〈Nc〉 grows nearly linearly with χ for ϕ = 0.72 (Fig. 4F) and is consistent with the observed slowing down of dynamics.

In addition to the least mobile particle clusters, one could equivalently look at the most mobile ones, which are also known to behave in a highly cooperative manner in dense liquids. The evolution of not just the size but also the nature of these dynamical heterogeneities with χ is notable for ϕ ≈ 0.85 (Fig. 5 and movie S7). In the enantiopure liquid (Fig. 5C), these heterogeneities were mostly large spinning vortices composed of almost 200 particles, with the chirality of these vortices being the same as the monomers’. At χ = 0 (Fig. 5A) and χ = 0.5 (Fig. 5B), we observed both (+) and (−) vortices, and these were also smaller in comparison to χ = 1. Besides vortices, we also observed streaming flows like those seen in liquids of achiral active particles (33). While orientational relaxation, which requires particles to rotate, is hindered at these ϕ values, these vortices and streaming flows, which have no counterparts in equilibrium liquids, provide new pathways (34) for translational relaxation of structure and is consistent with the observed weak growth in

$ταT$

.

Acknowledgments: We thank S. Gokhale for critical feedback on our manuscript. Funding: A.K.S. thanks the Department of Science and Technology (DST), Government of India, for a Year of Science Fellowship. R.G. thanks DST-SwarnaJayanti fellowship (2016–2021) for financial support. Author contributions: P.A. and R.G. conceived the research and designed the experiments. P.A. performed the experiments and carried out data analysis. A.K.S. contributed to project development and provided inputs with the manuscript. R.G. steered the research and wrote the paper with P.A. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.