Activity waves and freestanding vortices in populations of subcritical Quincke rollers

Significance

Active matter is made of units that move or displace others by using energy stored internally or gathered from their environment. In most systems and models considered so far, these self-propelled units are constantly moving. Here we study active matter made of units that do not move when isolated, but can be set into motion by close neighbors. Our subcritical active matter consists of Quincke rollers, that is, colloidal spheres at the bottom of a cell filled with conducting fluid submitted to a vertical electric field. We find spectacular collective self-organized phenomena: activity waves propagating in a quiescent population, and arbitrarily large, steadily rotating vortices forming without confinement or particle chirality.

Abstract

Virtually all of the many active matter systems studied so far are made of units (biofilaments, cells, colloidal particles, robots, animals, etc.) that move even when they are alone or isolated. Their collective properties continue to fascinate, and we now understand better how they are unique to the bulk transduction of energy into work. Here we demonstrate that systems in which isolated but potentially active particles do not move can exhibit specific and remarkable collective properties. Combining experiments, theory, and numerical simulations, we show that such subcritical active matter can be realized with Quincke rollers, that is, dielectric colloidal particles immersed in a conducting fluid subjected to a vertical DC electric field. Working below the threshold field value marking the onset of motion for a single colloid, we find fast activity waves, reminiscent of excitable systems, and stable, arbitrarily large self-standing vortices made of thousands of particles moving at the same speed. Our theoretical model accounts for these phenomena and shows how they can arise in the absence of confining boundaries and individual chirality. We argue that our findings imply that a faithful description of the collective properties of Quincke rollers need to consider the fluid surrounding particles.

Active matter nowadays designate situations in which energy is transduced in the bulk, at the level of individual units, into mechanical work (1, 2). Understandably, this definition engulfs a wealth of living systems, from mixtures of subcellular biofilaments and molecular proteins (3–8), to groups of cells (9–11), bacteria (12–17), and animals (18–22), but also nonliving ones such as autophoretic and other active colloids (23), shaken granular particles (24, 25), and robot swarms (26). Beyond the fascination continuously exerted by their often spectacular collective behavior—notably collective motion—there has been, over the years, significant progress in our theoretical understanding of the underlying physics (27).

The large majority of systems and models studied so far consist of interacting units that are constantly submitted to self-propulsion forces. This self-propulsion is considered intrinsic: A single active particle moves by itself. In this work, we explore a different situation in which a particle can only be set into motion when other particles, either by fluctuations or by their own self-propulsion, come close enough.

We performed experiments on Quincke rollers, that is, dielectric colloidal particles resting on a bottom surface, immersed in a weakly conducting fluid (28). When a steady uniform electric field $E$ is applied vertically across the experimental cell, colloidal particles acquire an antiparallel electric dipole. A small perturbation leads to a finite in-plane dipole component (perpendicular to the electric field) which then produces a net electrostatic torque (Fig. 1 A, Bottom). When the amplitude of the field exceeds a threshold value $E_{\mathrm{c}}$, this torque overcomes the viscous torque, triggering the rotation of particles. As the instability sets in, particles start rolling on the bottom surface at a constant speed $v_0$ along a randomly chosen direction. This is a supercritical instability, and one observes, as predicted, that $v_0^2\approx E^2-E_{\mathrm{c}}^2$ when $E>E_{\mathrm{c}}$. Quincke rollers are thus relatively simple self-propelled particles moving fast in two dimensions. Their collective properties in the supercritical regime $E>E_{\mathrm{c}}$ have been studied by the Bartolo group (29–32) and by others (33, 34), uncovering spectacular large-scale flocking phenomena that have been often argued to be well described by the Toner and Tu hydrodynamic theory developed in the context of simple dry models of locally aligning polar particles such as the Vicsek model (35).

Fig. 1.

(A) Schematics of the experimental setup. (B–J) Various snapshots with superimposed velocities. (Scale bar, 100 μm.) Colors indicate speed; quiescent particles are left white. A−E and F−J respectively show high and low area fraction cases (B–E, $\mathrm{\Phi }\simeq 0.4$; F–J, $\mathrm{\Phi }\simeq 0.2$) arranged by increasing values of $E/E_{\mathrm{c}}$ from left to right. (B and F) Quiescent clusters with local hexatic order obtained at $E=0.65\hspace{0.17em}{E}_{\mathrm{c}}$. (C) Wide activity wave at $E=0.89\hspace{0.17em}{E}_{\mathrm{c}}$. (D) Large-scale subcritical flock at $E=0.98\hspace{0.17em}{E}_{\mathrm{c}}$. (E) Subcritical flocking at $E=0.98\hspace{0.17em}{E}_{\mathrm{c}}$. (G) Localized activity at $E=0.9\hspace{0.17em}{E}_{\mathrm{c}}$. (H) Freestanding vortex at $E=0.948\hspace{0.17em}{E}_{\mathrm{c}}$. (I) Freestanding donut at $E=0.968\hspace{0.17em}{E}_{\mathrm{c}}$. (J) Spiral flock at $E=0.98\hspace{0.17em}{E}_{\mathrm{c}}$.

As a matter of fact, recent works demonstrated that the Quincke instability can set in below $E_{\mathrm{c}}$ when more than one particle are considered (36, 37). In a nutshell, the instability arises all the earlier (at lower field values) as the interparticle distance is smaller. Building on these local observations, we show here that Quincke rollers also exhibit striking collective phenomena in the subcritical regime $E<E_{\mathrm{c}}$. We find fast excitation waves, but also long-lasting, well-formed freestanding vortices, which are neither induced by boundary flow nor by some intrinsic chirality of particles.

We first present our experimental results, before explaining the mechanisms and interactions at play, which cannot be limited to a simple local alignment competing with noise. We build a mathematical model incorporating these ingredients and show that it can account for all phenomena observed. Our modeling effort indicates that the fluid surrounding rollers plays a key role, at odds with the conclusions of previous works in the supercritical regime. In particular, the collective motion taking place in our self-standing vortices arises from the combination of local subcriticality, activity, and hydrodynamics.

Experimental Results

Our experimental setup is similar to that used by the Bartolo group (29–32). It is briefly described in Materials and Methods and schematized in Fig. 1A. We use polystyrene spheres of diameter $9.9\hspace{0.17em}\mathrm{\mu }\mathrm{m}$ that rest and move at the bottom of a thin cell (horizontal size $2\times 2\hspace{0.17em}{\mathrm{c}\mathrm{m}}^2$, thickness $110\pm 10-\mathrm{\mu }\mathrm{m}$) filled with a $0.15\hspace{0.17em}\mathrm{m}\mathrm{o}\mathrm{l}/\mathrm{L}$ sodium bis (2-ethylhexyl) sulfosuccinate (AOT)/hexadecane solution. Typically, we found that $E_{\mathrm{c}}\approx 2.5\hspace{0.17em}\mathrm{V}/\mathrm{\mu }\mathrm{m}$. Our main control parameters are $\mathrm{\Phi }$, the global area fraction covered by particles, and the field intensity $E$. It is, in fact, hard to insure the quantitative reproducibility of observations: Under strong fields such as those employed, the electrodes deteriorate after a few tens of minutes, which allows for many measurements given the time scales involved, but not long enough to cumulate sufficiently many data points to build good statistics or a phase diagram. We have thus used multiple cells. From cell to cell, we observed variations in $E_{\mathrm{c}}$ of up to 20%. All results below are presented as a function of the relative position of $E$ with respect to $E_{\mathrm{c}}$, with $E_{\mathrm{c}}$ measured on each cell used.

The calculations of refs. 36 and 37, performed for particles suspended in the liquid, show that the closer the neighboring particles, the stronger subcritical effects, and, in particular, the lower the field value at which the Quincke instability can take place. Preliminary observations in ref. 38 indicated that this is also true for particles on a surface. It is thus desirable to create initial configurations with relatively dense areas. To obtain sufficiently high packing fractions, at least locally, we make use of the well-known effective attraction present well below $E_{\mathrm{c}}$, that has been used to study crystallization (39, 40). Indeed, the presence of particles near the bottom electrode distorts the local electric field, giving it a tangential component which induces an electrohydrodynamic fluid flow directed toward the particles. This fluid flow tends to bring particles closer to each other—an effective attraction which varies like $E^2$ (41).

In practice, we switch the field value to $E$ much lower than $E_{\mathrm{c}}$ (typically $0.6E_{\mathrm{c}}$), either from $E=0$ or from some supercritical regime with $E>E_{\mathrm{c}}$, and let the system relax under this effective attraction. If coming from a supercritical regime, particles first stop rolling. At low or moderate global area fraction $\mathrm{\Phi }$ (say $\mathrm{\Phi }<0.3$), one observes then the slow formation of almost crystalline clusters, possibly forming a percolating network in some denser areas (Fig. 1F and Movie S1). For rather large global area fractions $\mathrm{\Phi }\gtrsim 0.3$, the process leads to quasi-homogeneous initial configurations containing large areas with slightly larger density (Fig. 1B and Movie S2).

Activity Waves.

Increasing the field strength to values close to but below threshold (typically $0.9E_{\mathrm{c}}$) on configurations prepared as above triggers the motion of the particles located in the densest areas (Fig. 1G).

At high global area fractions, moving particles form fast solitary activity waves propagating in an otherwise quiescent system. These waves span large areas: They can be hundreds of micrometers long, and their width typically extends over dozens of active particles (Figs. 1C and 2A and Movie S3). Typical asymmetric density and speed profiles are shown in Fig. 2B. The wave speed and density profiles are fairly steady in time (compare the equally spaced profiles in Fig. 2B), but can vary from wave to wave, even within the same experiment. In particular, they increase with the density of the quiescent regions ahead. Of course, waves stop and disappear when reaching too sparse areas.

Fig. 2.

Activity wave in a dense sample ($\mathrm{\Phi }=0.4$, $E/E_{\mathrm{c}}=0.89$). (A) Snapshot of a portion of the wave ($800\times 500\hspace{0.17em}\mathrm{\mu }{\mathrm{m}}^2$ field of view, velocities indicated by arrows colored by speed, immobile particles not shown). (B) Profiles of speed and density (along the direction of propagation $X$, averaged over $Y$) at four different times separated by 0.1 s. (C) Time series of single particle quantities during the passage of the wave: (Top) displacement along $X$; (Middle) local density measured from inverse of Voronoi cell around the particle; (Bottom) particle speed.

Quiescent particles are recruited into the moving wave as its denser front reaches them. They then roll for, typically, tens of microseconds, over a distance of four to eight diameters, before stopping at the trailing edge, leaving a slightly sparser medium behind (Fig. 2B). The typical speed profile of a given particle entering and leaving the wave shows that the peak speed of particles in the wave is generally lower than $1\hspace{0.17em}\mathrm{\mu }\mathrm{m}/\mathrm{m}\mathrm{s}$, which is typically 3 to 4 times slower than the wave speed (Fig. 2 B and C).

These waves are reminiscent of those observed in classic excitable systems (42). They appear similar to the density waves observed during the aggregation stage of Dictyostelium discoideum (43, 44), but here no chemical signaling takes place: Our waves are pure “quorum-sensing” waves arising from transient motion of particles occurring when neighbors get close enough to trigger the Quincke instability.

Freestanding Vortices.

For moderate global area fraction ($\mathrm{\Phi }\approx 0.2$), our initial configurations consist of dense clusters, with some of them connected in a network. Increasing the field strength (again, to, typically, $0.9\hspace{0.17em}{E}_{\mathrm{c}}$), activity waves are triggered but now only propagate on this patchy landscape. If some particles become active in an isolated dense cluster, activity is short lived, with all particles in the cluster briefly set into motion, so that the cluster remains, slightly modified and sparser (Movie S4).

Activity triggered along the connected dense network can propagate continuously, possibly coming back repeatedly to the same location, gradually remodeling the network. These gradual changes of the network may allow the connection of heretofore isolated particles and clusters, recruiting more and more particles in the intermittently active population (Movie S5).

Often, the dense network forms loops or slightly larger areas such that the propagating wave “closes” on itself, nucleating a rotating packet. This rotating packet typically incorporates more and more incoming particles. Eventually, a large isolated vortex in which all particles rotate steadily is formed (Figs. 1H and 3 A–D and Movies S6 and S7), with few motionless particles surrounding it. Vortices of a variety of sizes are observed. When formed, they grow until no active particles reach them anymore. A vortex can also be perturbed by an incoming active cluster that may eventually destroy it. Apart from such events, vortices rotate as long as one experiment can last.

Fig. 3.

(A–D) Snapshots of a vortex under formation ($\mathrm{\Phi }=0.13$, $E/E_{\mathrm{c}}=0.9$, time indicated in yellow, field of view $240\times 240\hspace{0.17em}\mathrm{\mu }{\mathrm{m}}^2$); on the last image, the vortex has reached its “final” size, and its center of mass remains almost fixed. (E) Probability distribution function of angle $\phi$ between the radial direction and particle velocity calculated over all particles in vortex. (F and G) Time-averaged radial profiles of particle speed $v$ (Top) and local area fraction $\mathrm{\Phi }$ (Bottom) in steady vortex and donut configurations. (F) Evolution of profiles with increasing $E/E_{\mathrm{c}}$. (G) Profiles of different vortices observed at $E/E_{\mathrm{c}}=0.85$.

The chirality of vortices is random: Both clockwise and counterclockwise vortices are observed in statistically equal number. Over all experiments where vortices formed, we counted 96 clockwise and 94 counterclockwise ones, seen from above. This is in agreement with the absence of intrinsic chirality in our setup. Notably, the Quincke particles used are spherical to a very good accuracy, and thus achiral. Vortices emerge from spontaneous chirality breaking. We believe this is facilitated by the hydrodynamic interactions at play (see Modeling).

Vortices do not show solid-body rotation. Rather, particles forming them all move along the azimuthal direction at approximately the same speed (Fig. 3 E and F). The local area fraction inside vortices is also nearly constant in their bulk, with a slightly denser core, and a sparser boundary layer (Fig. 3F).

Steady vortices observed at the same field value (in various locations of the system) can take rather different sizes. They all share the same “bulk” properties, except for the smallest ones, for which the core and the outer boundary layer are not separated enough (Fig. 3G). There seems to be a minimal size for vortices, but not maximal one. We have observed vortices comprising several tens of thousands of particles.

Increasing the field strength, vortices first develop a sparser and slower core. They then become donut shaped, with a large inner empty region (Figs. 1 I and 3F and Movie S8). Increasing $E$ further, donuts are unstable and break into “flocking spirals” (Fig. 1J and Movie S9).

Phase Diagram.

From the several series of experiments performed, rescaling field values by the observed $E_{\mathrm{c}}$ for each sample, we obtain the rough phase diagram presented in Fig. 4, where the packing fraction $\mathrm{\Phi }$ is that observed in the field of view and averaged over the observation period (see Materials and Methods). For $E\lesssim 0.8\hspace{0.17em}{E}_{\mathrm{c}}$, all activity eventually dies out, leaving more or less connected dense crystalline clusters, depending on density. The various dynamic regimes described above are observed for $0.8E_{\mathrm{c}}\lesssim E<E_{\mathrm{c}}$, with $\mathrm{\Phi }\approx 0.3$ separating activity waves from vortices and donuts. More precisely, the transition separating quiescent asymptotic configurations from regime with sustained activity (within the experimentally accessible time) occurs at field values that decrease with increasing packing fraction. If this transition is likely to be an interesting absorbing phase transition (see Discussion), we stress that the dashed line separating vortices and donuts from activity waves is just a guide to the eye, since it is currently impossible, experimentally, to study the limit of stability of vortex and donut configurations. Finally, at field values very near the single-particle onset ($E/E_{\mathrm{c}}\approx 0.98$), we observe subcritical “flocking” regimes in which most particles are active and rolling. At low packing fraction, these flocks are typically elongated but turn frequently and may spiral in and out of themselves (Fig. 1J and Movie S9), whereas, at high density, they cover most of the space (Fig. 1 D and E and Movie S10).

Fig. 4.

Phase diagram of subcritical Quincke rollers in the $\left(E/E_{\mathrm{c}},\mathrm{\Phi }\right)$ plane. The area fraction $\mathrm{\Phi }$ is measured within the $1.7\times 1.4\hspace{0.17em}{\mathrm{m}\mathrm{m}}^2$ field of view. Phase labels were attributed by visual inspection of movies. In particular, the “vortex/donut” label was given as soon as one such steady structure was observed, irrespective of whether they coexist with other phenomena. Thus the dashed lines separating the active phases are mostly guides to the eye.

Modeling.

A precise mathematical account of Quincke rollers involves rather complicated interplay between three-dimensional (3D) flows, electrodynamics, and mechanics. Most models of their collective behavior are much simpler 2D effective descriptions that have been designed to deal with supercritical regimes. They fall into the dry polar active matter class (35), where self-propelled particles locally align their velocities, as in the Vicsek model. While this is justified by some calculations and general considerations, and proven a posteriori to provide a framework accounting satisfactorily for the phenomena observed (29–32, 34), models and continuous theories in the Vicsek class cannot describe the subcritical case. First of all, one would need to endow them with stop-and-go mechanisms characteristic of subcritical regimes: Isolated particles are immobile, but may start to roll when surrounded by close neighbors. Such an extension would probably account for the activity waves described above, but not for the spontaneous chirality symmetry breaking exhibited by freestanding vortices. Below, we build, from experimental observations, a wet hybrid model in which particles evolve in a 2D Stokes fluid and show that it accounts semiquantitatively for all collective phenomena observed.

We first review the mechanisms and interactions at play. Quincke particles interact via the 3D, mostly vertical electric dipoles they carry when submitted to a vertical electric field. This gives rise to some medium-range repulsion in the plane of motion that comes in addition to the hard-core repulsion due to their physical size. When particles are rolling, their dipole is slightly tilted, inducing a horizontal component opposite to the direction of motion (Fig. 1A). In addition, an effective attraction arises, due to electrohydrodynamic effects creating 3D flows directed toward particles (45). Rather far below $E_{\mathrm{c}}$, where no particle rolls, this leads to locally crystalline clusters (see, e.g., Fig. 1 B and F).

As $E$ is brought closer to $E_{\mathrm{c}}$, other interactions and mechanisms arise, which can be traced back mainly to the fact that the Quincke instability can be triggered at some density-dependent subcritical values of $E$. We, indeed, observe that dense clusters of nearby particles can spontaneously start moving (Movie S11A). Note that it is very difficult to define an effective threshold in such a case because it depends on the details of the local configuration (36), and experiments have imperfections. Along with density-dependent onset of motion, we also observe a density-dependent probability to stop. In particular, particles of a moving group that have separated themselves from neighbors stop earlier than those staying in the group (Movie S11B). Collisions between moving and quiet particles also provide stop and go mechanisms: Moving particles can “bump” into immobile ones, which often induces the motion of some of the particles hit. When an immobile particle is set in motion by a moving particle or a group of moving particles, it typically starts moving in the same direction as the incoming particles, as if pushed (Movie S11B).

Hybrid Model: Point Particles in a Continuous Stokes Fluid.

Based on the above experimental observations, we consider the following model where point particles $i=1,\dots ,N$, located at ${\mathbf{r}}_i$, interact locally and evolve in a 2D Stokes fluid on which they collectively exert some forcing, producing a velocity field $\mathbf{v}\left(\mathbf{r}\right)$ that advects and rotates them.

A key feature of the model is that each particle is endowed with a stochastic activity variable $a_i\in \left\{\mathrm{0,1}\right\}$. Inactive particles ($a_i=0$) do not roll and do not force the fluid. Active particles ($a_i=1$) move and entrain the fluid. As observed experimentally, inactive particles have a larger probability to become active if they have close neighbors, and all the more so if some of these neighbors are active. We thus generate a weighted local density variable $m_i=w_0{n}_i^0+w_1{n}_i^1$ where $n_i^{\mathrm{0,1}}$ are the numbers of inactive and active particles within distance $d_0$ of particle $i$, and $w_{\mathrm{0,1}}$ are some weights with $w_1>w_0$. A particle is activated or deactivated depending on the relative position of $m_i$ to a threshold value $m_{\mathrm{c}}$: When $m_i>m_{\mathrm{c}}$, an inactive particle can be activated with rate $k_{0\to 1}\left(m_i-m_{\mathrm{c}}\right)$, while active particles are deactivated with a rate $k_{1\to 0}\left(m_{\mathrm{c}}-m_i\right)$ when $m_i<m_{\mathrm{c}}$. Note that, when $m_{\mathrm{c}}\to 0$, all particles become almost unconditionally active.

When active (i.e., rolling), particles carry an orientation ${\theta }_i$ corresponding to the horizontal component of their tilted electric dipole. It is essentially enslaved to the rolling direction, but is also rotated by the local vorticity of the flow, and subjected to some angular noise as for active Brownian particles. The equations governing ${\mathbf{r}}_i$ and ${\theta }_i$ read

$${\dot{\mathbf{r}}}_i={\mathbf{u}}_i+a_i{\eta }_r{\mathit{\xi }}_i\hspace{1em}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\hspace{1em}{\mathbf{u}}_i=a_i\mathbf{v}\left({\mathbf{r}}_i\right)+{⟨\mathbf{F}\left({\mathbf{d}}_{ij}\right)⟩}_{j\sim i}$$ [1]

$${\dot{\theta }}_i=\left[\kappa \hspace{0.17em}\mathbf{e}\left({\theta }_i\right)\times {\mathbf{u}}_i+c\nabla \times \mathbf{v}\left({\mathbf{r}}_i\right)\right]\cdot \widehat{\mathbf{z}}+{\eta }_{\theta }{\xi }_i,$$ [2]

where ${\mathbf{d}}_{ij}={\mathbf{r}}_j-{\mathbf{r}}_i$, $\mathbf{e}\left({\theta }_i\right)$ is the unit vector along ${\theta }_i$, $\widehat{\mathbf{z}}$ is a unit vector perpendicular to the plane of motion, neighbors $j$ are the particles within distance $d_0$ of $i$, ${\mathit{\xi }}_i$ is a random-orientation vector of modulus drawn at each time from a unit-variance Gaussian distribution, and ${\xi }_i$ is a unit-variance Gaussian noise.

The only direct interaction between particles is via the pairwise repulsive force $\mathbf{F}\left({\mathbf{d}}_{ij}\right)$ composed of strong repulsion up to some core distance $d_{\mathrm{h}\mathrm{c}}<d_0$, followed by a weaker one accounting for dipole−dipole interaction cut off beyond $d_0$ for simplicity (Materials and Methods).

Particles collectively exert a force field $\mathbf{f}\left(\mathbf{r}\right)$ on the fluid. This force field and the fluid flow $\mathbf{v}\left(\mathbf{r}\right)$, like our entire model, are 2D, whereas the full Quincke system is 3D, even though the motion of particles takes place in a plane. This implies that our 2D effective fluid cannot be considered fully incompressible since it represents a part of the 3D (incompressible) flows of the full system. We thus decompose the velocity field into a compressible and an incompressible part, $\mathbf{v}\left(\mathbf{r}\right)={\mathbf{v}}_C\left(\mathbf{r}\right)+{\mathbf{v}}_I\left(\mathbf{r}\right)$, which are solutions of

$$\mathrm{\mu }{\nabla }^2{\mathbf{v}}_C\left(\mathbf{r}\right)-\alpha {\mathbf{v}}_C\left(\mathbf{r}\right)+\left(1-\epsilon \right)\mathbf{f}\left(\mathbf{r}\right)=0$$ [3]

$$\mathrm{\mu }{\nabla }^2{\mathbf{v}}_I\left(\mathbf{r}\right)-\alpha {\mathbf{v}}_I\left(\mathbf{r}\right)+\epsilon \hspace{0.17em}\mathbf{f}\left(\mathbf{r}\right)+\nabla p=0,$$ [4]

where $\mathrm{\mu }$ is the effective viscosity of the fluid, $\alpha$ is the effective friction with the substrate, $\epsilon$ is the fraction of the entraining force balanced by the incompressible part of the fluid, and $p$ is the in-plane pressure arising from the incompressibility condition $\nabla \cdot {\mathbf{v}}_I\left(\mathbf{r}\right)=0$.

The force field acting on the fluid consists of two terms,

$$\mathbf{f}\left(\mathbf{r}\right)=e{\sum }_i{a}_i\delta \left(\mathbf{r}-{\mathbf{r}}_i\right)\mathbf{e}\left({\theta }_i\right)+\lambda \nabla \rho \left(\mathbf{r}\right).$$ [5]

The first and dominant one is the entrainment of the fluid by active rolling particles. The second, which involves the density field $\rho \left(\mathbf{r}\right)={\sum }_i\delta \left(\mathbf{r}-{\mathbf{r}}_i\right)$, accounts for the flows induced by 3D electrohydrodynamics locally moving toward particles (45) (see Materials and Methods). This produces the effective attraction leading to crystalline clusters in the quiescent regimes.

This closes our equations. A few comments are in order. Note first that, in Eq. 1, the direction of motion ${\mathbf{u}}_i$, for simplicity, does not contain an explicit self-propulsion force, whose role is played here by the direct polar forcing of the flow in Eq. 5. Next, in Eq. 2, the first term aligns ${\theta }_i$ with the direction of motion, that is, essentially, the flow direction for an active particle. (Since active particles roll on the substrate [with friction], their rolling direction tends to align with the flow.) The second term is the classic Jeffery term involving the vorticity of the flow. Note that ${\theta }_i$ is irrelevant for inactive particles, since they do not force the fluid. Upon reactivation, ${\theta }_i$ needs to be reset. We found that assigning then the current direction of the repulsive forces ${⟨\mathbf{F}\left({\mathbf{d}}_{ij}\right)⟩}_{j\sim i}$ produces dynamics very close to the experimental observations, although other choices are possible and work well too.

Modeling Results.

Simulations of the above model are not hard to conduct (see Materials and Methods for details). It is relatively easy to find parameter values such that one of the phenomena described above emerges. As a matter of fact, many features of the model are not necessary to account for one particular type of dynamics (say, activity waves), but they are all useful for comprehensive and faithful results. One key difficulty is that the model has no explicit electric field parameter, and thus building a phase diagram similar to that in Fig. 4 is not obvious. The activity threshold $m_{\mathrm{c}}$ is a key parameter, since $E\to E_{\mathrm{c}}$ implies, by construction, that $m_{\mathrm{c}}\to 0$. However, there is, a priori, no reason for $m_{\mathrm{c}}$ to be the only parameter changing when $E$ is varied.

We inspected model parameters, fixing first those whose typical value can be read from experimental data, determining next those whose value does not matter much, and those that are important for the emergence of activity waves, freestanding vortices, etc. (see Materials and Methods). We found that, because these phenomena occur in a rather limited range of field values ($E/E_{\mathrm{c}}\in \left[0.8;1\right]$), most parameters can, in fact, be considered constant, except $m_{\mathrm{c}}$ and $e$, the strength of entrainment of the fluid by active particles (compare Eq. 5). We scanned the $\left(m_{\mathrm{c}},e\right)$ parameter plane at various packing fractions $\mathrm{\Phi }$ (with all other parameters fixed at reasonable values determined beforehand). From this, we found that varying $m_{\mathrm{c}}$ in the range $\left[12;0\right]$ and taking $e=e_0+e_1\exp \left[-m_{\mathrm{c}}/m_0\right]$ with $e_0=e_1=0.15$ and $m_0=0.5$ accounts very well for varying $E/E_c$ within the range $\left[0.6;1\right]$ in experiments. (Details about this procedure can be found in SI Appendix.) The resulting phase diagram in the $\left(\mathrm{\Phi },m_{\mathrm{c}}\right)$ plane is shown in Fig. 5J. Semiquantitative agreement with Fig. 4 is achieved, with the same phenomena occurring in about the same regions of parameter space. (Note that a linear variation of $e$ with $m_{\mathrm{c}}$ works almost as well, indicating the robustness of results.)

Fig. 5.

Modeling results. (A–E) Typical configurations in a domain of linear size $960\hspace{0.17em}\mathrm{\mu }$m with periodic boundary conditions. Disks of diameter $d_{\mathrm{h}\mathrm{c}}=10\hspace{0.17em}\mathrm{\mu }$m are centered on particle’s positions. Immobile particles are in white, while moving ones are colored by their speed (colormap in E). (A) Activity waves propagating from left to right in a dense system ($m_{\mathrm{c}}=8.9$, $\mathrm{\Phi }=0.33$, about 3,900 particles). (B) Subcritical flocking regime in a dense system with particles mostly moving from right to left ($m_{\mathrm{c}}=3$, $\mathrm{\Phi }=0.47$, about 5,530 particles). (C–E) Vortex and donut formation ($\mathrm{\Phi }=0.12$, 1,433 particles). (C) Snapshot during transient ($m_{\mathrm{c}}=3$). (D) Final vortex comprising most particles in the system ($m_{\mathrm{c}}=0.5$). (E) Final donut configuration comprising most particles in the system ($m_{\mathrm{c}}=0.05$). (F) Speed (Top) and density (Bottom) profiles at four equally separated times of one of the waves present in A. (G) Time series of single particle quantities (averaged over 10 particles) during the passage of the activity wave presented in F: (Top) displacement along $X$; (Middle) local density measured from inverse of Voronoi cell around the particle; (Bottom) particle speed. (H and I) Time-averaged radial profiles of particle speed $v$ (Top) and local area fraction (Bottom) in steady vortex and donut configurations. (H) Evolution of profiles with increasing $m_{\mathrm{c}}$. (I) Profiles of different vortices observed at $m_{\mathrm{c}}=3$. (J) Phase diagram in the $\left(m_{\mathrm{c}},\mathrm{\Phi }\right)$ plane, obtained from simulations in the same domain as in A–E.

This agreement extends to all measurements that have been performed experimentally. Here we only present results corresponding to the observations and measurements reported in Figs. 1–3. (A more complete study of the model will be presented elsewhere.) Snapshots of activity waves, freestanding vortices and donuts, and subcritical flocking regimes are shown in Fig. 5 A–E (see also Movies S12–S16).

The structure and dynamics of activity waves produced by the model (Fig. 5A and Movie S12) are similar to the experimental observations: Both their typical static and temporal profiles (Fig. 5 F and G) are in good agreement with those of Fig. 2. Like in experiments, the width and speed of activity waves can vary. In particular, they are larger when propagating in denser quiescent regions.

The model easily produces freestanding vortices and donuts (Fig. 5 D and E). Like in experiments, their size varies from run to run, and even within a single run, depending on the “history” of incoming particles. Sometimes, small vortices can be destroyed by incoming active packets, events also present in the real system. Fig. 5C shows a typical snapshot during the transient leading to stabilized vortices. The model does not incorporate any mechanism explicitly breaking chiral symmetry. Unsurprisingly, thus, clockwise and counterclockwise vortices are observed in statistically equal number, like in experiments. The inner structure of the model vortices is similar to that of experimental vortices: Particles inside move overwhelmingly along the azimuthal direction; in the bulk, particles roughly move at constant speed, and the density is nearly constant (Fig. 5 H and I).

An important question is whether vortices of any size can form, or whether, on the contrary, there is a length scale limiting their maximal size. Experimentally, the vortices observed depend on the “history” of the system, and we do not have much control over their size, so that we cannot answer this question. In the model, on the other hand, we can engineer vortices at will. Starting from an initial dense blob of particles in a large domain, a vortex quickly forms. We can then gradually feed the vortex with more and more particles. Following this protocol, we have been able to produce a vortex comprising 45,000 particles, and we are confident that even larger vortices are stable. This huge vortex is represented in Fig. 6 in a composite representation showing both particle and flow velocity.

Fig. 6.

Very large self-standing vortex obtained from simulations of the model at $m_{\mathrm{c}}=4$, with ∼45,000 particles rotating clockwise. In the four quadrants of this snapshot, we represented fluid flow speed $\left|\mathbf{v}\right|$ (Top Left), fluid velocity orientation $\arg \left(\mathbf{v}\right)$ (Top Right), and particles colored by their speed (Bottom Right) and by their orientation ${\theta }_i$ (Bottom Left). The domain represented has a linear size of 2,600 μm.

Discussion

We have shown that collections of Quincke rollers can exhibit nontrivial emergent dynamical phenomena in the subcritical regime where one isolated particle remains immobile. Detailed experimental observations and our modeling have revealed that one key feature of this subcriticality is the stop-and-go mechanisms which govern the activation by neighbors and the spontaneous deactivation of particles via some effective local density. Some initial activity can thus propagate in space and time, for some time or forever, depending on parameters, notably, the packing fraction of particles and the strength of the electric field applied. We have uncovered two main phenomena with sustained activity: waves and freestanding vortices.

In dense enough subcritical Quincke systems, activity propagates in waves traveling in an otherwise quiescent medium. These waves are reminiscent of those present in excitable systems, but here the propagation mechanism is “purely mechanical,” a situation somewhat similar to what happens in some models of so-called mechanical waves in cell tissues (11, 46–48).

Even though freestanding vortices are well known to occur in animal groups such as fish schools or ants (see, e.g., refs. 49–51), they are rather rare in better-controlled active matter systems. In many published works, single vortices emerge among interacting self-propelled units because of confinement (24, 25, 32, 52–60). Another frequent case is that of particles with some intrinsic chirality or which spontaneously break chirality. One then typically observes many vortices with a well-defined size, sometimes arranged regularly in space (4, 61–63).

Unconfined vortices have been reported before in achiral active colloids systems driven by electric or magnetic fields. Rather complex 3D, toroidal, vortical structures were found in refs. 64–66. Two-dimensional freestanding vortices exist in magnetically driven rollers, but they seem somewhat unstable, and they may be limited in size (67, 68). Freestanding vortices have even been shown to exist in a standard Quincke roller system, but one submitted to a pulsating electric field, yielding rotating clusters with complex internal structure that seem to have a well-defined typical size (33). From this review, we conclude that the subcritical Quincke roller vortices described and modeled here are possibly the first occurrence of arbitrarily large, 2D, “fluid” freestanding vortices in achiral active matter.

Modeling Remarks.

Interacting Quincke rollers actually constitute a complicated system with many different effects at play, such as 3D electrohydrodynamic flows. More generally, hydrodynamic forces are often recognized to play a major role for collections of colloidal particles activated by external fields (69, 70). It may thus come as a surprise that most if not all works modeling collective properties of supercritical Quincke rollers make use of the 2D Toner and Tu equations, with some satisfactory agreement with experiments (29, 30, 34, 71). Indeed, the Toner−Tu equations constitute the hydrodynamic theory of the dry polar active matter class whose best-known particle-level representative is the Vicsek model (35).

For the subcritical Quincke rollers studied here, such a framework cannot be satisfactory, even if the Vicsek particles are supplemented by repulsive interactions, something suggested to be necessary to account for the confined vortices studied in ref. 32. The surrounding fluid must be involved in a faithful model, not for the emergence of activity waves—they can probably emerge in a dry aligning model with stop-and-go mechanisms—but to account for the formation of freestanding vortices such as those reported in this work and, we believe, in the pulsating forcing case of ref. 33. The step-by-step construction and rather systematic exploration of our model has shown that the stability, regularity, and arbitrary size of self-standing vortices require, in particular, the partial incompressibility of the 2D fluid (with a fully incompressible fluid, vortices are unstable) and the “attractive flows” modeled by the $\lambda$ term in Eq. 5 (the usual vorticity Jeffery term in Eq. 2 does not suffice).

The 2D model proposed here offers a framework for the study of emergent properties of Quincke rollers. It accounts for 3D mechanisms that are essential in the subcritical regime, such as the electrodynamically induced weak attraction between particles. It generically leads to some effective alignment of particles via the surrounding fluid, without the need for explicit, Vicsek-style alignment. Even though the surrounding fluid is incorporated, such a wet active matter model remains numerically very efficient, allowing for easy simulation of millions of particles.

Perspectives.

We conclude by offering two important points worth studying in the future.

Whereas we have demonstrated that our model accounts very well for the subcritical phenomena reported here, we believe it could well be a better description of the supercritical regimes than the simple Toner−Tu/Vicsek framework. We note, in passing, a very recent work suggesting that systems such as those made of Quincke rollers must have hydrodynamic descriptions fundamentally different from the Toner and Tu equations (72).

Another point of interest is the nature of the transition separating sustained from transient activity, that is, the line delimiting the blue region in Figs. 4 and 5J. To its left, any initial activity eventually dies out, leaving an absorbing state. Infinitely many such absorbing states exist. In a dry setting, the transition should fall in the conserved directed percolation class, also observed in so-called random organization phenomena, and the absorbing states reached on the transition line should be hyperuniform (73–81). What is the universality class of this phase transition for the Quincke rollers studied here, and, more generally, in the context of wet subcritical active matter, is an important problem. Unfortunately, this is currently very difficult to study experimentally due to the difficulties in preparing samples with identical properties (see Materials and Methods). On the other hand, this problem could be tackled within the simple model presented here.

Materials and Methods

Experimental Setup.

Colloidal particles (Thermo Scientific G1000, polystyrene spheres) with diameter $9.9\hspace{0.17em}\mathrm{\mu }\mathrm{m}$ are dispersed in a $0.15\hspace{0.17em}\mathrm{m}\mathrm{o}\mathrm{l}/\mathrm{L}$ AOT/hexadecane solution. The suspension is sealed in a $2\times 2\hspace{0.17em}{\mathrm{c}\mathrm{m}}^2$ cell constructed by two indium-tin-oxide–coated glass slides which are separated by $110\pm 10-\mathrm{\mu }\mathrm{m}$-thick insulating glass spacer (Fig. 1 A, Top). All particles sediment as a monolayer on the bottom electrode. The electric field is generated by a voltage amplifier.

To obtain reasonable control on the water component, we first dry AOT at $80°$C for 10 h to remove the water completely. The AOT/hexane mixture is then prepared at a room temperature of $25°$C and a humidity of 40%. Experiments are then conducted under the same conditions. Before dispersing colloidal particles, the AOT/hexane mixture is maintained for 24 h to reach equilibrium.

To follow and investigate the motion of individual particles, their dynamics are observed with a Zeiss microscope (10× objective) and recorded by a complementary metal-oxide semiconductor camera at a rate of 500 frames per second. Using IDL software, the positions of particles are located and tracked with an accuracy of $140\hspace{0.17em}\mathrm{n}\mathrm{m}$ (0.1 pixel).

At the field strength typically applied ($E\approx 2\hspace{0.17em}\mathrm{V}/\mathrm{\mu }\mathrm{m}$), the cell can work continuously for less than 1 h typically, after which the conducting film of the electrodes is too damaged, and particles get stuck on the surface.

In this system, the global area fraction is not uniform. This is true especially near the boundary of the cell where the moving particles tend to accumulate. The observations presented here are all made in the central region of the cell. The global area fraction $\mathrm{\Phi }$ reported in this study is that measured within the field of view, which is $1.7\times 1.4\hspace{0.17em}{\mathrm{m}\mathrm{m}}^2$.

Pairwise Repulsion Force.

The intensity of the pairwise repulsion $\mathbf{F}\left(\mathbf{d}\right)$ in Eq. 1 is a continuous function of $d=\left|\mathbf{d}\right|$ composed of a stiff harmonic part followed by a screened dipole−dipole term: $F\left(d\right)=k_{\mathrm{h}\mathrm{c}}\left(d-d_{\mathrm{h}\mathrm{c}}\right)+f\left(d_{\mathrm{h}\mathrm{c}}\right)-f\left(d_0\right)$ for $d<d_{\mathrm{h}\mathrm{c}}$, and $F\left(d\right)=f\left(d\right)-f\left(d_0\right)$ for $d>d_{\mathrm{h}\mathrm{c}}$, where $f\left(r\right)=-p^2\partial r\left[r^{-3}\exp \left(-r/d_{\mathrm{s}\mathrm{c}\mathrm{r}}\right)\right]$. For simplicity, the angular part of the dipolar interaction $f\left(r\right)$ has been omitted, because it is rather weak compared with the repulsive one. (We checked that dipole alignment is not crucial for the emergence of vortices or activity waves.)

Electrohydrodynamic Forcing of the Fluid.

The term $\lambda \nabla \rho \left(\mathbf{r}\right)$ in Eq. 5 is obtained by considering all particles surrounding a given position r, say at a small distance $ϵ$. Electrohydrodynamics produces an axisymmetric 3D flow around each of them that locally goes toward the particle (45). One can thus write the total forcing of the flow at position r as $\oint d\phi \hspace{0.17em}\rho \left(\mathbf{r}+ϵ\mathbf{e}\left(\phi \right)\right)\hspace{0.17em}{f}_0\mathbf{e}\left(\phi \right)\simeq ϵf_0\oint d\phi \hspace{0.17em}\left(\mathbf{e}\left(\phi \right)\cdot \nabla \right)\rho \left(\mathbf{r}\right)\mathbf{e}\left(\phi \right)\simeq \pi ϵf_0\nabla \rho \left(\mathbf{r}\right)$.

Determining Model Parameters.

The unit length was set to be the particle diameter $d_{\mathrm{h}\mathrm{c}}=1$, corresponding to about $10\hspace{0.17em}\mathrm{\mu }\mathrm{m}$. We found that $d_{scr}=0.8$ and $p=0.3$ are appropriate values for the parameters involved in the pairwise repulsion acting beyond $d_{\mathrm{h}\mathrm{c}}$. The overall interaction distance, for simplicity, was fixed to $d_0=2$ (a value beyond which the repulsion is very small, and thus negligible).

The parameters ruling the stop-and-go mechanisms were fixed at simple values in the absence of precise experimental estimates: $w_0=1=\frac{1}{2}{w}_1$ and $k_{0\to 1}=k_{1\to 0}=1$. This, together with our choice $d_0=2$ (within which neighbors are counted), leads to having the range $m_{\mathrm{c}}\in \left[0;12\right]$ roughly correspond to the experimental range ($E/E_{\mathrm{c}}\in \left[0.6;1\right]$).

In Eqs. 1 and 2, the noise strengths ${\eta }_r$ and ${\eta }_{\theta }$ should not be taken too large (otherwise, as expected, most patterns are destroyed). In fact, these noises can be considered residual, and they just help “regularize” some local dynamical events. We worked with ${\eta }_r=0.04$ and ${\eta }_{\theta }=0.01$, but these numerical values are not crucial in any way. In Eq. 2, the Jeffery coefficient in front of the fluid vorticity was not taken to its classic value $c=1/2$; $c=1$ was found to provide more regular vortices. This leads also to a larger “vortices” region in the model phase diagram, in line with the experimental phase diagram. (Note that, in our 2D system, the fluid is an effective representation of complicated 3D flows, so that $c$ need not be equal to 1/2. Moreover, local vorticity not only rotates the colloids but, at the same time, may modify the structure of ionic double layer, so that the colloid rotation is more susceptible to flows.) Finally, we worked with $\kappa =2.5$, a value of the “rolling-alignment” parameter large enough to ensure the rapid enslaving of ${\theta }_i$ to the direction of motion ${\mathbf{u}}_i$.

The basic fluid parameters were set to $\alpha =\mathrm{\mu }=1$, yielding a hydrodynamic screening length $\sqrt{\mathrm{\mu }/\alpha }$ of the order of $10\hspace{0.17em}\mathrm{\mu }\mathrm{m}$. We found that this rather short range is preferable in order to observe frequent enough nucleation of vortices. Larger values still allow for stable vortices of about any size, but they do not emerge as easily. The fraction of effective incompressible fluid should not be too large; otherwise, activity waves are not well defined. Self-standing vortices, on the other hand, are quite stable even with incompressible 2D flows. Results presented are at $\epsilon =1/3$. The electrohydrodynamic flow strength $\lambda$ controls the strength of the effective attraction between particles, and thus the typical density of near crystalline clusters forming in the quiescent states. We found that $\lambda =1/3$ yields interparticle distances in clusters similar to those observed in experiments.

Finally, unit time was set to correspond to 1 ms so that the typical speed of active particles is of the order of $1\hspace{0.17em}\mathrm{\mu }\mathrm{m}/\mathrm{s}$, as observed in the experiments. (Note that the speed of active particles is essentially set by the entrainment parameter $e$.)

Details of Numerical Simulations.

The model does not present any significant numerical difficulty. We used simple Euler time stepping with a time step of $0.01\hspace{0.17em}\mathrm{m}\mathrm{s}$. At each time step, the first task is to update $a_i$, the activity index of particles, according to the activation/deactivation rates and the state of current neighbors of each particle. Next, the density field $\rho \left(\mathbf{r}\right)$ and the force field $f\left(\mathbf{r}\right)$ (Eq. 5) are constructed by averaging particles’ positions and orientations over square boxes of linear size $10\hspace{0.17em}\mathrm{\mu }\mathrm{m}$. These coarse-grained fields, thus defined on a square lattice, are then Fourier transformed so that the Stokes Eqs. 3 and 4 can be solved easily. Next, the resulting velocity field is transformed back, yielding the real-space field $\mathbf{v}\left(\mathbf{r}\right)$. This velocity field is finally used to update the positions and orientations of particles following Eqs. 1 and 2, where the other interactions and noises are also taken into account.

Data Availability

All study data are included in the article, SI Appendix, and Movies S1−S17