Playing with active matter

Active-matter systems consist of “active particles,” which are living or synthetic agents that generate mechanical forces resulting in self-propulsion.¹ Some typical examples of active motion include the swimming behavior of Escherichia coli bacteria and the self-propulsion of artificial Janus particles.^2,3

Active-matter systems can be of very different sizes, ranging from microscopic colloidal systems^1,4 to groups of animals or robots and even human crowds. In spite of their seemingly large differences, all these systems show similar emergent behaviors. Important examples are the motion observed in flocking birds, the formation of living crystals made of active particles,⁴ the motility-induced phase separation seen in suspensions of self-propelled hard spheres,⁵ and the chaotic swarming observed in swimming bacteria.

Active motion leads to a steady dissipation of energy into the environment. Therefore, systems of active particles are not in thermodynamic equilibrium. Active particles can dynamically self-organize and exhibit collective behaviors that are impossible at thermodynamic equilibrium.¹ However, the interest for active matter does not only stem from its statistical physics properties. It is a field that might also benefit technological applications, from delivering drugs targeting organs, to controlling the spread of infectious microorganisms,⁶ and developing microrobots capable of advanced group behaviors.^7,8

Most active-matter experiments are, however, performed with microscopic particles and require advanced microfabrication and microscopy techniques.⁹ In this article, we introduce a simple macroscopic experimental model for active matter based on Hexbugs, which are small robots that self-propel in a random direction.¹⁰ After showing that this motion can be described as an active Brownian motion, we demonstrate how it can be turned into a chiral active Brownian motion by modifying the robots. Then, we make the Hexbugs interact with different obstacles. We show how they can propel and rotate some simple passive objects. We also make them interact with complex environments, where they are sorted based on their motility and chirality. Finally, we demonstrate the emergence of Casimir-like activity-induced attraction between planar objects in the presence of active particles in the environment.

A single active particle moving in a two-dimensional space has a position given by the coordinates $[x(t),y(t)]$⁠. Its self-propulsion results in motion with speed v along a direction given by the particle orientation $φ(t)$ (defined with respect to the x-axis). As it moves on a horizontal surface, the particle undergoes rotational diffusion with rotational diffusion coefficient $DR$¹¹ due to the microscopic irregularities of the surface. In other words, it experiences erratic and random changes in its orientation. An example of this motion is shown for a Hexbug Nano robot in Video 1 of the supplementary material.

A Hexbug (Fig. 1(a)) has six curved rubber legs on each side and a vibrational motor. When turned on, the vibration of the motor propagates to the legs. The resulting friction between the legs and the surface causes the robot to move forward with a constant speed (for a detailed model of the Hexbug propulsion mechanism, see Appendix A in the supplementary material; for a discussion of the conditions under which it can be considered as an active particle, see Sec. VI). Due to noise and small imperfections in both the terrain and the robot, the orientation of the Hexbug changes over time, leading to rotational diffusion.

To record its motion, we allowed a Hexbug to move within a $44×44 cm2$ arena bounded by some cardboard walls, as shown in Fig. 1(b). This and other details of the experimental setup are given in Appendix B in the supplementary material. We placed a camera above the table to film the Hexbug in motion. We glued a bright-colored tag to the robot and tracked it using the OpenCV library (this library is available for Python and C++¹²).

In Subsections II A and II B, we characterize the motion of a single Hexbug and show that it is in fact an active Brownian motion. Then, we show how a Hexbug can be modified to perform a chiral active Brownian motion. Before proceeding further, we note that the dynamics of such self-propelling extended objects is more complex than that of an active Brownian particle. In fact, Hexbugs are finite-sized, elongated, macroscopic objects taking their momentum from their interaction with the ground. This implies a coupling between their rotational and translational degrees of freedom,¹³ differently from what happens in an active Brownian particle, where no such coupling is present. The coupling is particularly important when the velocity of the Hexbug is not aligned with its orientation, e.g., after collisions. In the following, we will neglect this coupling and its effect on the dynamics of the Hexbugs, for the sake of simplicity and to emphasize general properties of active matter that are common to micro- and macroscale.

We can transform a Hexbug into a chiral active Brownian particle by gluing a small 3D-printed parallelepiped on top of the robot (see Appendix B for specifications in the supplementary material). If this object is glued off the Hexbug's axis of symmetry, its weight continuously exerts a torque that bends the trajectory. For example, when the weight is placed on the Hexbug starboard (right-hand) side, as shown in Fig. 1(d), the robot tends to turn right. This leads to a right-chiral active Brownian motion. The resulting motion is shown in Fig. 1(e) (Video 2 in the supplementary material). By placing the weight on the Hexbug port (left-hand) side (Fig. 1(g)), the robot tends to turn left, leading to a left-chiral active Brownian motion (Fig. 1(h) and Video 3 in the supplementary material). The impact of the added weight on the motion chirality might slightly vary for each Hexbug due to the variability of the Hexbug geometry, but this can be compensated by finely tuning the exact position of the weight.

We now consider the effect of the interaction of active particles with movable objects and fixed obstacles. In the case of movable objects, the presence of active particles generates an active bath that can displace them. If these objects are asymmetric, they can in turn induce a directional motion or rotation in the active particles. In the same way, fixed obstacles can be used to alter the way in which the active particles explore the space available to them. To create the objects and obstacles used in these experiments, we glued together small 3D-printed cylinders (⁠ $radius=1.9 cm$⁠, $height=2 cm$⁠) so that we could control their shape and dimension (see Appendix B in the supplementary material).

When an active particle encounters a movable object, it can interact with it, for instance, by pushing it in its direction of motion. If the object is asymmetric, it will be propelled because the active particles interact with it asymmetrically. For example, when we placed a movable wedge in the arena with 14 Hexbugs (Fig. 2(a) and Video 4 in the supplementary material; the Hexbugs were randomly placed in the arena one by one, see Appendix B), we observed that the Hexbugs tended to get trapped on the wedge's concave side and push it. Meanwhile, Hexbugs tended to interact for much shorter times with the wedge's convex side, because they slid along the wedge without getting trapped. We tracked the tip of the wedge and measured its total displacement as a function of time (Fig. 2(b)). It grows approximately linearly in time at a rate of $≈0.038±0.05 cm s−1$⁠.

In the case of a linear rod (Fig. 2(c) and Video 5 in the supplementary material), the Hexbugs interacted with both sides of the rod in an equal manner, resulting overall in much less movement. In fact, when the Hexbugs enter into contact with the rod, they align with it and slide along its length without pushing it too much. As a consequence, the rod's displacement growth rate, shown in Fig. 2(d), is about one order of magnitude smaller than the wedge's.

In the literature, there are several observations of passive objects interacting with an active bath. Several experiments showed that symmetric objects such as spheres or rods immersed in an active bath feature superdiffusion at short time scales and enhanced diffusion at long time scales.^‡‡18 There are also several experiments with asymmetric objects propelled in an active bath. In particular, Kaise et al.¹⁹ demonstrated how the activity of motile bacteria in a solution pushed a wedge similarly to what we have shown in Fig. 2(a) and Video 4 in the supplementary material. All these phenomena are studied because they demonstrate a mechanism through which the active system provides energy that is successfully converted into mechanical work and can be used to propel microscopic objects. This is appealing because (1) the active system is inherently chaotic and (2) the possibility to propel microscopic asymmetric shapes paves the way toward biomedical applications (e.g., targeted drug delivery to organs).

We can also consider the case where there are fixed obstacles in the environment. Because active particles are not in thermodynamic equilibrium with their environment, it is possible to use the features of the environment to perform complex tasks on the active particles, e.g., disposing fixed obstacles to trap, or sort them based on their motion properties.¹ 

We performed an experiment with a series of wedges fixed along the middle of the arena, with a gap between them of approximately 2 cm, wide enough for the Hexbugs to go through. We then placed 14 Hexbugs (one by one and randomly distributed, see Appendix B in the supplementary material) in the arena and let them move freely. As time passes (see Video 6 in the supplementary material), these wedge-shaped barriers effectively trap the active particles on the left side of the arena (highlighted by the red solid line in Fig. 2(e)). This is expected because the concave shape of the barrier rectifies the motion of the Hexbugs and either makes them turn around or traps them in its corners. Figure 2(f) shows the fraction $ρL$ of active particles in the left portion of the box as a function of time. With our system parameters, the distribution quickly approaches a plateau of $ρL≈0.9$⁠.^§§

Also in this case, we performed a control experiment using rods instead of wedges as fixed obstacles (Fig. 2(g)). In this case, the Hexbugs remain uniformly distributed in the arena (see Video 7 in the supplementary material). This is quantified by measuring the fraction of active particles on the left side of the arena (area highlighted by the red solid line in Fig. 2(g)). As shown in Fig. 2(h), the distribution remains at $ρL≈0.5$ for the entire duration of the experiment.^***

Similar behaviors have been observed with microscopic particles. For example, Galajda et al.²⁰ showed that E. coli bacteria can be concentrated by a wall of funnels to the subspace to where the funnel openings lead (like those in Fig. 2(e)). They also repeated the experiment with motile bacteria using a flat wall with openings (as in Fig. 2(g)) finding that, in this case, the concentration remains the same, on average, in the two subspaces. This effect was reproduced with simulations in Ref. 21. This is important for several reasons, for example, it (1) suggests an operative way to discriminate active particles from passive ones and (2) inspires methods to sort microorganisms belonging to the same species based on their motility, which can be useful in biology.

In this section and Subsection III D, we deal with gears and their interaction with the Hexbugs. Gears are mechanical components that transmit rotation and power from one shaft to another and are used in almost all machines and appliances. The point of these two subsections is to show that, from a set of active particles that move independently from one another and have no evident collective rotation behavior, one can manage to extract a well-defined, steady rotation, i.e., how the disordered translational kinetic energy of the collection of Hexbugs can be converted into rotational energy. This conversion is made possible thanks to the design of the shape of the gear, which must have a defined chirality (this sets the direction of the rotation) and appropriate dimensions.

To demonstrate that it is also possible to transfer torque to objects, we placed movable asymmetric gears (see Fig. 3(a)) in the arena. The gear we consider has multiple concave corners created by their teeth. When active particles interact with these gears, they behave similarly as in the case of the V-shaped objects. However, in this case, the propulsion of the Hexbugs also causes the gear to rotate.

We placed the gear in the middle of an arena with 14 Hexbugs (Fig. 3(b) and Video 8 in the supplementary material). We tracked the motion of the gear by placing two bright-colored tags on top of it: one on the center and the other on one tooth. The active particles tend to get trapped in between the saw teeth and push the gear. Because the saw teeth are located all around the gear, the propelling force also creates a torque, so that the gear also rotates in a direction that depends on the orientation of its teeth. To quantify this behavior, we measured the cumulative angle of rotation, i.e., the sum of the gear's orientation change in each time step (Fig. 3(c)). As expected, the data show that the gear rotates in the left-chiral direction (Fig. 3(d)). The rate of the rotation is $≈+0.3±0.2 rad s−1$⁠. We expect this value depends on the mass of the gear, more massive gears holding smaller values of angular rotations for the same experimental conditions, and on the geometry of the gear's teeth.

We ran a second experiment with a right-chiral gear of roughly the same size and mass (Fig. 3(e) and Video 8 in the supplementary material). The same video analysis determined that the behavior of the active particles was similar. However, the gear rotates in the right-chiral direction (Fig. 3(f)). The rate of the rotation is $≈−0.6±0.2 rad s−1$⁠. The difference between the left-chiral and right-chiral rates of rotation can be ascribed to the following facts: (1) in different experiments, the Hexbugs runs different trajectories, and therefore the number of collisions with the gear is different; (2) the two gears used in the experiments were built with different materials (plastics of different density) and their mass, moment of inertia, and interaction with the surface were different. This affects how much energy is transferred and converted each time to motion and rotation and how much is lost into friction, during a collision event with a Hexbug.

These behaviors were first predicted with numerical simulations, then experimentally realized using microscopic gears set in active baths of motile Bacillus subtilis²² and E. coli²³ bacteria. The dimensions of the gears used in the experiments were in the range of tens to hundreds of micrometers and the achieved angular speeds of the order of a few revolutions per minute (rpm). Interestingly, these works demonstrated that it is in principle possible to extract energy from the disordered motion of an active bath through directed rotational motion. There have also been alternative proposals to obtain directed rotational motion by using catalytic self-propelling Janus particles²⁴ and by generating the torque using thermocapillary forces²⁵ or self-electrophoresis.²⁶ 

We finally explore the behavior of the system when the gears are fixed to a surface. To achieve this, we placed the hollow left-chiral gear on top of a metallic nut glued at the center of the arena (Fig. 3(g)). This setup restrains the position of the gear while still enabling it to rotate. We let 14 Hexbugs move freely within the arena (Fig. 3(h) and Video 10 in the supplementary material). We observe that the Hexbugs get trapped in the teeth of the gear, inducing its right-chiral rotation, with a rotation rate $≈+1.1±0.2 rad s−1$ (Fig. 3(i)). The rotation rate is enhanced when the gear is fixed to the surface. Because of the presence of the pivot, now the whole magnitude of the force exerted by a Hexbug on the gear contributes to the rotational motion only, differently from the case of a movable gear, where the force contributes to both the rotational and translational motion.

To compare, we performed the same experiment with the right-chiral gear (Fig. 3(j) and Video 11 in the supplementary material). As expected, the behavior was similar, but displaying right-chiral enhanced rotation (Fig. 3(h)) at a rate of approximately $≈−0.7±0.2 rad s−1$ (Fig. 3(f)).

A microscopic version of this experiment was performed in Ref. 27, where motile bacteria were used to power an array of fixed microgears. The angular speed of each microgear depended on several factors, but it could reach up to $20 rpm$ with an edge speed very close to the speed of a freely swimming bacterium. This result showed that it is possible to exploit motile bacteria to rotate fixed microgears, like microscopic draught animal moving equally microscopic millstones.

In this section, we show that, by modifying the environment appropriately, we can sort chiral active particles according to their chirality. To achieve sorting, we assemble a structure called a “chiral flower.” This structure is made up of a set of fixed ellipses placed equidistantly along a circle to resemble flower petals that are tilted with respect to the radial direction.²⁸ For this experiment, we employ an arena of $88×44 cm2$ with two mirror-symmetric chiral flowers. We built these chiral flowers by 3D-printing 12 blue and 12 yellow oval-shaped obstacles (the petals of the chiral flowers) and by placing them along two circles of inner radius $10 cm$ with a tilt of +60° and −60°, respectively (Fig. 4(a)). We modified two Hexbugs to have right chirality (Fig. 1(d)) and two Hexbugs to have left chirality (Fig. 1(g)). We then released the four Hexbugs in the arena.

Figures 4(c)–4(e) show a series of snapshots of the motion of a right-chiral Hexbug (violet trajectory) in a right-chiral flower: the bug remains trapped within the flower until a left-chiral Hexbug (orange trajectory) enters the right-chiral flower and disrupts its motion, pushing it out of the right-chiral flower. The left-chiral Hexbug itself quickly moves out of the right-chiral flower because its motion is not matched to the geometry of the right-chiral flower.

We quantify how many times each Hexbug is found within a specific flower by recording the experiment and analyzing each snapshot. The snapshots are spaced equally in time so that counting the number of frames each Hexbug is inside a specific flower is equivalent to measuring the time spent in each of them (for the experiment, see Video 12 in the supplementary material). Figure 4(b) shows that the left-chiral Hexbugs were found in the left-chiral flower seven times more frequently than the right-chiral ones and that the right-chiral Hexbugs were found two times more frequently in the right-chiral flower than the left-chiral ones.

Looking at Fig. 4(b), one might wonder whether the right flower is less selective than the left flower in trapping the chiral Hexbugs, despite the two flowers having been designed to be nominally mirror-symmetric. The observed discrepancy can be due to several factors. One could be a simple statistical factor: if the two chiral flowers are equivalent, the discrepancy must become smaller with an increased number of repetitions of the experiment. Moreover, small differences in the placement of the petals and the variability observed in the Hexbugs regarding their chirality tendency to rotate $Ω$ and self-propulsion speed v might have made the chiral flowers not equivalent in trapping. Last but not least, the Hexbugs are solid objects and interact with one another so that collisions between Hexbugs can alter their theoretical trajectories, e.g., by freeing a trapped Hexbug as shown in Figs. 4(c)–4(e).

Mijalkov and Volpe²⁸ showed via numerical simulations that chiral flowers can sort active particles of different chiralities down to the size of large biomolecules. We remark that our experiment slightly differs from these simulations, where the mutual interaction between chiral active particles was neglected. In our experiment, Hexbugs are macroscopic, elongated objects that, when simultaneously present in the arena, cannot occupy the same volume so that their interactions cannot be neglected. This becomes evident when performing the sorting experiment using several Hexbugs at the same time. We decided to show the movies with four Hexbugs because it was the optimal way to demonstrate the effect in a clear way.

We finally qualitatively demonstrate how movable objects in an active bath can experience some activity-induced attraction. In fact, when the persistence length of the active particles is comparable to the characteristic size of the confining geometry, their intrinsic active nature can give rise to an attractive force between the movable objects.³⁰ This is reminiscent of the Casimir attraction experienced by parallel metallic plates set at a small distance, as a consequence of the confinement of the fluctuations of the vacuum electromagnetic field.³¹ Another example of fluctuation-induced forces is the critical Casimir force experienced by colloidal particles in a binary critical mixture close to a second-order phase transition due to density concentration fluctuations.^32,33

To observe this effect, we place two parallel, straight rods at a distance of about $15 cm$ in the arena where 14 Hexbugs are present. The distance at which the rods are placed initially has been chosen in such a way that they are far enough from each other and from the boundaries. In Fig. 5(a) and Video 13 in the supplementary material, we see how, as time passes, the rods are pushed closer to each other.

Although there is no model we could compare our results quantitatively to, similar behaviors have been observed with microscopic particles. Angelani et al.³⁵ demonstrated the attractive action of a bacterial bath on a solution of suspended spherical particles with experiments and numerical simulations. Ray et al.²⁹ simulated the attraction between parallel plates caused by active particles. Using a minimal model for the active particles and their interaction with the plates, they showed that the effective (attractive) force between the plates increases with increasing running length of the active particles.^†††

To go beyond this simple qualitative experiment, one could get inspiration from the existing literature on the topic. Ni et al.³⁶ show that the effective interaction between two parallel hard walls in a 2D suspension of self-propelled (active) colloidal hard spheres can be tuned from a long-range repulsion to a long-range attraction by changing the density of active particles. Kjeldbjerg and Brady³⁰ provide a theory for the interaction between parallel plates mediated by active particles. It is worth noting that both Refs. 29 and 30 assume a parallel plane geometry and infinitesimally small particles, while in our experiment the parallelism of the rods is not enforced and the size of the active particles is finite. Harder et al.³⁷ investigated the role of the shape of the suspended objects in determining the forces mediated by active particles, assuming finite-size active colloids; they found that large, elongated objects, like our rods, typically interact attractively, as we observed in our experiment.

In the experiments we have presented, the Hexbugs emulate the behavior of active particles. These small robots can be used to demonstrate nonequilibrium phenomena at a macroscopic scale, suitable for the classroom. In fact, performing physical and tangible experiments, as opposed to using only simulations,¹⁴ makes the learning process more intuitive.

As Hexbugs are macroscopic objects of finite size, their dynamics are more complex than those of an active Brownian particle, because of the coupling between their angular and the translational degrees of freedom.¹³ Furthermore, inertia might induce delays between their orientation and velocity³⁸ and alter their dynamics in the presence of an external field acting on the same plane of their self-propulsion. For this reason, all our experiments are performed on a horizontal, flat surface, where inertial effects can be effectively neglected, to keep the focus on the features that are common to microscopic and macroscopic active matter.

Beyond the examples presented in this article, it is possible to use the Hexbugs to emulate many other experiments that have been performed with active matter, such as those described in Ref. 1. For example, it is possible to further modify the environment to create other interesting behaviors, e.g., by adding objects or obstacles of different sizes and shapes, or by letting them run over a concave parabolic substrate, such as in Ref. 39, which, combined with gravity, mimic a harmonic trap. Another interesting set of experiments is reported in Ref. 40, where Hexbugs are employed to model gas pressure and electrical conduction. Complex dynamics due to inertial and effect are reported in Ref. 41. It is also possible to modify the Hexbugs to make them respond to some external stimuli, like light^42,43 or sound intensity.^44–48 The Hexbugs can also be used to reproduce the transition between free flow and a congested state in traffic,⁴⁹ the sorting of chiral particles by polarized wall currents,⁵⁰ and to build flexibles and self-propelled superstructures capable perform tasks such as infiltrating narrow spaces and move around obstacles as in Refs. 51 and 52.

See the supplementary material for guidelines for the experiments, description of the experimental videos, a brief discussion of the meaning of superdiffusion and enhanced diffusion, and three figures illustrating the fitting of the averaged trajectory of the Hexbugs to extract the parameters of their active motion.

The authors gratefully acknowledge the support of the following funding: Grants ERS–StG No. 677511 “COMPLEXSWIMMERS” and ERS–CoG No. 101001267 “Microscopic Active Particle with Embodied Intelligence” of the European Research Council, Grant No. 2019.0079 of the Knut and Alice Wallenberg Foundation, and the EC MSCA–ITN Grant No 812780 “ActiveMatter” of the European Union's Horizon 2020 research and innovation programme.

The authors have no conflicts to disclose.

The Hexbug Nano is a small toy robot manufactured by Innovation First, Inc. (length $4.4 cm$⁠, width $1.2 cm$⁠, height $1.7 cm$⁠, mass $7 g$⁠). The legs of the Hexbug account for $1.0 cm$ of its total height. Every Hexbug has six pairs of legs. The legs are angled and rubbery. This causes the mechanical energy of the vibrations to move the robot forward.

The total force that the Hexbug exerts on the substrate has not only a normal component (to the substrate) but also a parallel component. In the model that we will present, we will consider only the parallel component, because the Hexbug has no vertical motion. The maximum amount of friction force that the surface can develop is greater during the first half of the cycle than during the second half. If the inclination of the legs and the motor induced-force are set properly, the horizontal backward component of the motor-induced force will not overcome the maximum amount of static friction developed by the surface, while the horizontal forward component of the motor-induced force will overcome it and the Hexbug will start moving forward.

From the previous discussion, the Hexbug motion is equivalent to having two different static friction coefficients for the forward (⁠ $μsbw$⁠) and backward (⁠ $μsbw$⁠) motions. The kinetic friction coefficient (⁠ $μk$⁠), however, is the same for both cases. Let the motor be described as an effective vibration motor with frequency $ν$ and period $T=1/ν$⁠. In the first half of one cycle (⁠ $t∈[0,T/2]$⁠), the motor pushes the Hexbug forward with a constant force F, in the second half of one cycle (⁠ $t∈[T/2,T]$⁠) the motor pushes the Hexbug backward with the same (in modulus) constant force F.

A “perfect” Hexbug moving on a “perfect” surface will keep a constant average speed and never change its direction. However, irregularities in the microscopic structure of the surface and in the Hexbug itself cause its orientation to change slightly over time. In Sec. II, we have shown that its motion can be described (on a horizontal surface) by Eqs. (1)–(3).

When a small load is added asymmetrically on the Hexbug, it has a tendency to preferentially turn toward that direction. In such a case, it behaves like a chiral active particle that can be described by Eqs. (2), (3), and (9).

As a final remark, in both Eqs. (1)–(3) and Eqs. (2), (3), and (9), noise affects the Hexbugs orientation, but not its position variables x and y, differently from the case of microscopic particles suspended in a fluid. This is because our system is macroscopic: the dimensions of the Hexbug, and the interaction with the surface and with the surrounding air are such that no relevant noise acts on the position. Instead, for a microscopic particle, the interactions with its fluid environment are such that diffusion due to the presence of white noise is observed also on the particle position.