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All issues Volume 3 / No 4 (2024) Natl Sci Open, 3 4 (2024) 20240005 Full HTML
Special Topic: Active Matter
Open Access
Review
Issue
Natl Sci Open
Volume 3, Number 4, 2024
Special Topic: Active Matter
Article Number 20240005
Number of page(s) 36
Section Physics
DOI https://doi.org/10.1360/nso/20240005
Published online 17 May 2024

© The Author(s) 2024. Published by Science Press and EDP Sciences.

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

INTRODUCTION

Active matter is a class of non-equilibrium systems, composed of self-propelled units, which can convert their own or ambient energy into mechanical energy required for the directed motion on an individual level. Active matter can be found across diverse spatial scales, ranging from meter-sized animals, centimeter-sized robots, micrometer-sized bacteria, active colloids, and nanometer-sized molecular motors [1, 2]. Due to its intrinsic far-from-equilibrium characteristics, active matter showcases a series of exotic phenomena and novel dynamics that are unprecedented in equilibrium state, attracting widespread attentions. The number of review articles, examining various aspects such as hydrodynamics, interaction with complex environments and other properties in active matter [1-10], has soared in the dynamic field. The research goal of the active matter is twofold: Firstly, it incorporates biological systems into the scope of soft condense matter physics, thereby opening up significant opportunities in the development of intelligent materials, biomedical and other areas. Secondly, it anticipates the emergence of new statistical and thermodynamic laws, and answers fundamental non-equilibrium issues.

As mentioned earlier, almost all living systems are affiliated with active matter, and thus a great deal of exciting progress has been made with utilizing living systems [3, 9, 11]. However, there are some disadvantages in the biologically active systems. For instance, controllable parameters of the constituent agents are often insufficient, and some of them can be challenging to control, such as individual mobility, status, and life cycles. This can result in poor experimental reproducibility and large errors. Besides, the decoupling of some inevitable biochemistry from physics poses great challenges for the elucidation and establishment of the theoretical models. To address the issues posed by biological systems, artificial active matter is becoming a promising candidate by virtue of uniform individual properties, multiple controllable parameters, convenient control, and the ability to decouple biochemical and physical processes. In the past two decades, researchers developed several experimental systems represented by active colloids, vibrating active particles, and robots to perform a myriad of investigations. Among these artificial active systems, active colloids of a mesoscopic size draw intensive attentions, and a series of review articles have also been published on the topic [12-15]. However, a comprehensive overview of macroscopic, artificial active matter remains absent, as shown in the green box of Figure 1.

thumbnail Figure 1

Active particles are macroscopic, microscopic and nanoscopic in size and have propulsion speeds (typically) up to a fraction of a centimeter per second. The letters correspond to the artificial particles in Table 1 [16-39]. The insets show examples of biological and artificial swimmers. Three representative particles of macroscopic artificial active matter are displayed in the green box: robots, Hexbugs, and surfers (Adapted with permission from Ref. [2]. Copyrightc2016, American Physical Society).

In this review, we aim to provide a comprehensive overview of recent advances in macroscopic artificial active matter, especially with a focus on experimental progress. Macroscopic artificial active matter not only shares the common advantages of artificial active particles but also possesses the extra benefits such as simple and economic manufacturing, scalability of motion control modules, flexibility of switching between hydrodynamic and dry environments, and convenience in experimental measurements. These characteristics enable them particularly well-suited for investigating collective behaviors in strongly constrained and crowded environments. The objectives of macroscopic artificial active matter are also dual: it aims to serve as a bionic experimental platform that mimics life systems by continuously introducing more complex interactions; on the other hand, it strives to retain and precisely control only the most critical parameters, allowing for a rigorous examination and exploration of the fundamental theory of active matter. In Section “An individual active particle: theoretical model and experiment”, we delve into the motion model of active particles. Considering scenarios where inertial and chiral effects play significant roles, the underdamped Langevin equation often proves to be a suitable tool for describing the motion of these macroscopic particles. We then elucidate the motion mechanisms of various active particles, along with their dynamic characteristics and statistics in both free and complex environments. Sections “Collective behaviors in vibration-activated particles” and “Robots in active matter” provide a comprehensive review of the collective behavior observed in “dry” active matter, with a specific focus on vibration-activated particles and robots. Here, we examine their structure, dynamic properties, and potential applications. Additionally, we briefly outline the progress made in modeling collective motion based on these experimental contributions. In Section “Macroscopic particles in a hydrodynamic environment”, we shift our focus to macroscopic active particles in hydrodynamic environments, compensating for the lack of hydrodynamic interplay observed in “dry” systems. Finally, in Section “Outlook and perspectives”, we offer a forward-looking perspective on the future development of macroscopic artificial active matter, highlighting emerging opportunities and challenges.

Table 1

Table 1Examples of experimentally realized artificial active particles. The letters in the first column correspond to the examples plotted in Figure 1 [2]

AN INDIVIDUAL ACTIVE PARTICLE: THEORETICAL MODEL AND EXPERIMENT

Models for the motion of a macroscopic particle

There are numerous well-established works on theoretical models for motion of active matter, and we will not redundantly elaborate on them. The following relevant references are recommended for further reading [1-3, 9, 40, 41]. As is well known, the Langevin equation is applicable to macroscopic artificial active particles. Nevertheless, unlike the microscopic and mesoscopic counterparts, the ratio of the mass of the particle to the environmental viscosity, in many macroscopic scenarios, is already a finite value, and thus the inertial effect cannot be ignored. As a result, the underdamped Langevin equation is more frequently applied in them,(1)(2)where m is the particle mass and J the moment of inertia. V0 is the self-propulsion speed of the active particle, M an effective torque. denotes a translational (rotational) friction coefficient. represents particle orientation , θ(t) is the angle of the particle orientation with the x-axis. is a random vector, whose components and are Gaussian random numbers with zero mean and variances representing white noise from the surrounding. , and , kBtt denoting an effective thermal energy quantifies the translational noise strength. Similarly, and , kBTR denoting an effective thermal energy quantifies the orientational noise strength. The symbol denotes a noise average. Based on the equations of motion (1) and (2), a series of independent quantities such as the long-time translational self-diffusion coefficient can be given by Ref. [42],(3)The superposition of the translational diffusion and an active term leads to a substantial enhancement of the effective diffusion coefficient. The O(D0, D1) is a corresponding dimensionless correction, and D0 and D1 are dimensionless delay numbers unique to the inertial system. For more analytical results such as the orientational correlation function or the equations of motion generalized to non-inertial frame, see the references [42-44].

The inertial effect strongly influences the kinematic characteristics of active particles [43, 45]. At times, the thermal noise associated with translation and rotation is significantly less than the self-propulsion force of active particles. In other words, the diffusion caused by the thermal bath is several orders of magnitude weaker than the displacement caused by self-propulsion. In this instance, if there is no other random noise present, we can safely disregard the noise term, and the self-propelled motion is linear along the particle orientation. The importance of inertia is clearly demonstrated in the experiment utilizing 3D-printed vibration-driven active particles, with a notable inertial delay observed between the orientation and velocity of the particles [44]. This inertial delay significantly impacts long-term diffusion, making it distinctly different from both passive particles and overdamped active particles.

When inertia is negligible, the Eqs. (1) and (2) simplify to an overdamped Langevin equation as shown below, which is typically employed to describe the motion of mesoscopic active chiral particles, including colloidal particles and bacteria.(4)(5)Or more intuitively, the equations can be restated in another form. For convenience, we only present the equations of motion in a two-dimensional environment. The models for active Brownian motion can be straightforwardly generalized to the case in three dimensions.(6)(7)(8)where x(t) and y(t) are the position coordinates of an active particle, () is the translational (rotational) diffusion coefficient of the active particle, and is the angular velocity of the particle. The physical interpretation of the other parameters remains consistent with that in Eqs. (1) and (2). For a chiral active particle, the mean square displacement (MSD) is obtained as [42](9)From the diffusive long-time limit for the MSD, e.g., , the long-time effective diffusion coefficient can be expressed by(10)It is worth mentioning that if the particle is a linear one, ω or M will vanish, the MSD can be analytically given by Ref. [46](11)Sometimes the persistent length and the persistent time replace the self-propulsion speed and rotational diffusion coefficient in these equations. Figure 2 presents the MSD curve measured from an experiment, clearly demonstrating the transition from superdiffusion at short time scales (MSD ) to normal diffusion at long time scales (MSD ).

thumbnail Figure 2

Mean square displacement of a macroscopic active particle with different self-propelled velocities. For a passive Brownian particle (v0=0 cm s-1), the motion is always normal diffusion [MSD ], while for an active particle, the motion is diffusive ballistic at intermediate time scales [MSD ], and again normal diffusion but with an enhanced diffusion coefficient at long time scales.

If an active particle is placed in a confined space, the MSD eventually approaches a saturated plateau due to the confinement effect. However, it is still possible to obtain the effective diffusion coefficient by linearly fitting the part of the MSD curve prior to saturation. Notably, the effective diffusion coefficient strongly depends on the geometric characteristics of the confined space, and can be derived from the Smoluchowski equation [47],(12)ω(x) is the width of the confined space. If the confined space is periodic, the Eq. (12) becomes,(13)The symbol indicates averaging over the periodic space.

Let’s consider the motion of an active particle on a two-dimensional unrestricted surface as an example. To document the particle’s trajectory, a camera is employed during the experiments. Subsequently, software tools are utilized to extract coordinates from the trajectory data. These coordinates are then used to compute the statistical quantities such as mean square displacement (MSD) for the translocation and rotation, further to obtain the self-propulsion speed, orientational correlation function, and translational (rotational) friction coefficient etc. For example, , Δφ measured from experiments represents the change in orientation angle over time Δt, and the notation indicates ensemble average.

Dynamics and statistics of an individual active particle

The exploration of a single particle, whether in a simple or complex environment, primarily focuses on three aspects: (1) the mechanisms behind the self-propulsion of the particle, (2) statistical properties such as velocity distribution and fluctuations, as well as diffusion behaviors that may range from sub-diffusion to normal diffusion, super-diffusion, or even the caging effect, and (3) the dynamics that arise from interactions with the external environment. For instance, appropriately established obstacles, boundaries, or potential fields can all steer the directional motion of an active particle, leading to intriguing phenomena like chiral separation, rectification, and Hall-like flow.

When it comes to macroscopic active particles whose self-propulsion is powered by external vibration, the mechanism of energy conversion heavily relies on the interplay between friction, inertia, and inelastic interactions between the particle and the vibrating boundary. Directed motion, on the other hand, often necessitates particles endowed with anisotropic properties, which can encompass shape, mass distribution, elasticity, and more. As a result, most active particles are dumbbells, rods, or other objects with linear asymmetry [48-50]. Additionally, the energy can also come from internal energy reserves, such as a battery that converts electrical energy into directed motion via a build-in motor. Owing to variations in energy sources and particle shapes, the mechanisms for self-propulsion are also diverse.

In their study for the mechanism of self-propulsion, Koumakis et al. [51] proposed a mechanical model that connects the geometric shape and kinetic properties of a 3D printed active particle to its self-propelled velocity when bouncing on a vibrating plate. They found that by tilted flexible legs, there is an optimal tilt angle which maximizes mobility. In addition to the translation mechanism, the mechanism of rotational motion of a chiral particle has also been experimentally and theoretically studied. Inspired by rattleback toys, a metal wire is bent to misalign the principal axes of its moment inertia tensor relative to the axis of curvature, enabling it to rotate in a preferred direction on a vertically vibrating table [52]. Scholz et al. [53] also studied the self-propulsion mechanism of a centimeter-sized rotor that can transform translation into rotation when placed on a vibrating plate. They determined that the interaction between the elastic deformation of the rotor’s legs and dynamic/static friction between the vibrating table and the rotor is the origin of its dynamics. They also revealed that various collisional rebound modes of the rotor result in diverse rotational behaviors, see Figure 3.

thumbnail Figure 3

Illustration for self-propelled mechanism of different particles (A) Chiral bent-wire (Adapted with permission from Ref. [52]. Copyright c 2005, American Physical Society). (B) Vibrating vibrot (Adapted with permission from Ref. [53]. Copyright c 2016, IOP Publishing Ltd. and Deutsche Physikalische Gesellschaft). (C) 3D printed walkers, in which structures in top row have connected (rigid) legs and structures in bottom row have free (flexible) legs (Adapted with permission from Ref. [51]. Copyright c 2016, IOP Publishing Ltd and Deutsche Physikalische Gesellschaft). (D) Self-propelled dimer made of two grain-filled balls. Upper panel, instantaneous horizontal velocity vs. time. Lower panel, vertical position of both balls and the vibrating plate as a function of time [57] (Adapted with permission from Ref. [57]. Copyright c 2017, Spring Nature).

Theoretically, it is predicted that connecting two spheres with a simple harmonic spring or rigid rod to form a dimer will produce a ratchet effect, or directional motion, if the internal degrees of freedom of the dimer are broken. This can be achieved through factors such as the difference in size of the two spheres [54], the different responses of the two spheres to the external potential field [55], and the different friction coefficients of the two spheres [56]. Inspired by this concept, macroscopic artificial self-propelled dimer can be manufactured by breaking the symmetry of energy dissipation. The active dimer consists of two ping-pong balls connected by a rigid rod, with each ball filled with granular beads of a different dissipation rate. When placed on a vibrating table, the dimer undergoes periodic collisions and rebound modes, converting vertical momentum into horizontal one [57].

Apart from the exploration of self-propelled mechanisms, an individual or several non-interacting particles are applied to investigate new statistical laws in out-of-equilibrium systems, as well as to research the fluctuation relation of active particles moving in a medium. Kumar et al. [58] experimentally conducted a statistical analysis of the velocity of a self-propelled rod immersed in passive particles and showed that it undergoes frequent steps in the opposite direction of its mean direction of spontaneous motion. The velocity distribution is highly non-Gaussian, and the large deviation function (LDF) exhibits a kink at zero velocity. The antisymmetric part of the LDF is linear, indicating that the velocity fluctuations obey a symmetrical relationship similar to the entropy production rate. Followed by their own work, they tested the validity of the isotropic fluctuation relation (IFR) using the same experimental system [59]. The statistical measurement of the velocity is largely described by the anisotropy generalization of the IFR. The fluctuation relation of the translational velocity of active particles in a passive bath has been clarified, but the question remains whether the fluctuation relation holds in an active bath. To answer the question, the probability density function (PDF) of angular velocity for the asymmetric rotor in a Hexbug (toy robots) bath has been measured, revealing a strong non-Gaussian profile. The symmetry in the PDF is related to the detailed fluctuation relation between entropy production [60]. When two vibration-activated rods in Ref. [51] move in a close-packed granular medium, a nonreciprocal pursuit-and-capture interaction between the two rods, similar to that between predators and prey, was observed. In the background medium, one rod adjusts its direction of motion, while the other rod maintains its original direction, eventually the two rods aligning and moving side by side. A coarse-grained theory was proposed to explain the intriguing effective attraction between two rods. In this model the rods can be viewed as a moving point-force density which gives rise to elastic strains in the dense medium, reorienting other rods. The x and y components of the displacement field U is given by Ref. [61],(14)(15)In most cases of active matter, the sources of activity and noise are usually distinct. However, for vibration-activated particles, the noise and activity arise from the same collisional forcing. Despite this unique correlation between noise and activity, an experimental investigation of individual particle motion within a vibrating granular monolayer confirmed that the ABP model is also applicable to vibrating particles. This conclusion was arrived at by comparing particle motion on both short- and long-time scales with the theoretical predictions of the model [62].

When active particles move in a complex environment, they often interact with the surroundings, leading to a range of unexpected dynamics. A prime example of this is the behavior of a Hexbug, a centimeter-scale toy robot, within a dish antenna, which functions as a harmonic trap, see Figure 4. Unlike conventional self-propelled particles, Hexbug exhibits a self-aligning property that results in the particle circling around the center of the trap. The self-alignment can be described by the overdamped equation , which incorporates the self-aligning torque with an orientation n towards a velocity v. By manipulating the persistence of the robot’s orientation, it is possible to switch between the “orbiting” and “climbing” state [63].

thumbnail Figure 4

(A) Hexbug running in a parabolic dish; (B) close-up picture of a Hexbug. n denotes the particle orientation. (c) Observed orbiting trajectory, the arrows represent the particle orientation n and self-propelled velocity v. The color codes are representative of the passage of time, aiding us in distinguishing and identifying the trajectories of the Hexbug over time (Adapted with permission from Ref. [63]. Copyright c 2019, American Physical Society).

Tapia-Ignacio et al. [64] introduced the surface roughness of the substrate to make more stochasticity into the Hexbug’s motion. Additionally, they examined the dynamical statistics of the robot within a similar harmonic trap, including diffusion coefficient, mean square velocity, radial and velocity probability distributions. The comparison between theory and experiment confirmed that the inertial effect should be considered in the model. Similarly, by placing Hexbug in a harmonic trap, confinement imposed by the potential well induces the accumulation of Hexbugs within a finite distance of the boundary, resulting in three distinct dynamical states. These states can be tuned by activity, and only occur in cases where inertia cannot be disregarded [65].

Active particles in most studies are generally found moving on a rigid substrate, but there is limited research on the motion of active particles on a highly deformable substrate. The authors theoretically and experimentally studied the orbital dynamics of a vehicle moving on an elastic membrane (as illustrated in Figure 5), which is affected by an external curvature field, as well as the dynamics of controlling multiple vehicles through local surface deformation [66].

thumbnail Figure 5

(A) One vehicle transiting around a central depression; (B) side view of the differential driven vehicle; (C) different k(r) functions for prograde (light blue) and retrograde (dark blue) precession measured from experiments. The decreasing k has the same trend as the measured tilt angle γ(r) (black stars) (Adapted with permission from Ref. [66]. Copyright c 2022, PNAS).

Besides the effects of the potential field or substrate on the dynamics, the interaction between particles and obstacles or other particles can lead to intriguing behaviors. Horvath et al. [67] discovered that a Hexbug can be compelled to move opposite to its internal propulsion when colliding with a moving wall in a one-dimensional channel. The switch from a forward mode to a backward mode is primarily due to the asymmetry caused by the Hexbug’s legs bending backwards. Additionally, researchers employed a 3D printed granular particle to explore the dynamics in a quasi-1D circular channel. They found that under environmental constraints, a chiral active Brownian particle (ABP) undergoes a transition to become a run-and-tumble particle (RTP). In this state, the RTP exhibits a distinct motion pattern: it moves steadily in a specific direction for a set duration, then transitions to erratic movement in place for a period, before resuming its steady motion in a new direction. This behavior closely resembles the movement patterns observed in motile organisms like E. Coli bacteria and Chamydomonas algae [68]. The researchers linked the switching of these two modes to the dynamics of a two-state molecular motor and simplified the influence of constraints to a periodic torque. This torque was mapped to the dynamics of a Brownian particle moving within a potential under a constant external force [69].

The interplay between active particles and environmental complexity can also lead to the realization of complex functions such as particle rectification, separation, and capture. The boundary, as a complex environment, is coupled with the chirality of Hexbugs, enabling chirality sorting [70]. In particular, when the collisional angle is optimal, the chirality causes the particle to rebound and then orbit back to collide with the boundary, resulting in ballistic transport along the boundary. Clockwise (counterclockwise) rotating Hexbugs will move directionally along the left (right) side of the boundary, leading to the selective collection of two types of particles. As another type of complex environment, obstacles can also be used to achieve the capture and sorting of active particles, as shown in Figure 6A [71].

thumbnail Figure 6

(A) (a) Top view of the sorting experiment with right-turning particles and left-turning particles; (b) particle extraction rate as a function of the total number of particles in the channel for the horizontal (α=0°) and the tilted experiments (α=1.8°); (c) particle locations in the channel without tilt; (d) particle locations in the channel with a tilt α=1.8° ((A) (a)–(d) Adapted from Ref. [70]. Copyright c 2020, American Physical Society). (B) Typical trapped and untrapped states in experiment and simulation (Adapted with permission from Ref. [71]. Copyright c 2019, American Physical Society).

Utilizing V-shaped structures, macro-sized active rods can be trapped and collected. The competition between the particle accumulation within V-shaped structures due to persistent motion and the particle escape along the inclined walls of the V-shaped structure is responsible for the particle capture. With the sensitivity of particle capture to the persistent length, the V-shaped structure can also be applied to separate particles with various activity (Figure 6B). The interaction between active particles and obstacles can also be applied for the separation of active chiral rotors [72]. The rotors consisting of two Hexbug robots arranged in antiparallel directions, traverse an array of fixed obstacles. After collision with the obstacles, the lateral migration direction depends on the chirality of the rotor. The rotational direction of the rotor and its frictional interaction with the obstacle determine the lateral migration direction of the rotor, resulting in chiral separation. In recent experiments on the interaction between active matter and obstacles, a self-deforming snake robot was used to investigate scattering dynamics. The experimental results show that when the robot interacts with a single obstacle, the scattering angle exhibits a prominent peak. However, when multiple obstacles are present, the shift of the scattering angle results in a secondary peak [73]. Finally, the asymmetry in response to external signals can lead to emergent behavior. A Kilibot robot carries a light sensor in an eccentric position, which is equivalent to be affected by a non-axisymmetric light field. The resulting trajectories are orbital, indicating the existence of bistability. This asymmetric response may provide insights into extraordinary behaviors of biological systems that interact with environmental signals [74].

COLLECTIVE BEHAVIORS IN VIBRATION-ACTIVATED PARTICLES

Active matter is a system that comprises active agents, where the interactions between individuals and the nonlinear interplay between particles and the environment can stimulate a more abundant macroscopic dynamics than that observed in equilibrium state through collective modes. These diverse phases and the associated collective dynamics demonstrate the profound charm of active matter physics, engaging the interest of a growing number of researchers.

Due to the relatively large particle size, typically in the millimeter and centimeter range, macroscopic, artificial active matter is generally confined within environments with boundaries, except for a few robotic experiments that do not impose boundaries or one-dimensional experiments that use ring-shaped boundaries to simulate a periodic boundary condition. Furthermore, the self-propulsion of most macroscopic artificial active particles does not require a hydrodynamic environment and is thus considered as a “dry” active system, where momentum is not conserved. It is worth noting that there are also examples of macroscopic artificial active matter in hydrodynamic environments, which will be introduced in the following section. Three types of active particles commonly adopted in experiments are vibration-activated particles, robots, and particles in hydrodynamic environments. In this section, we focus on the progress of collective behaviors of vibration-activated particles.

Statistics in collective motion

An intriguing and distinctive statistical feature of active particles is the number fluctuation. By vibrating nonpolar rods, a huge, long-lived number fluctuation has been experimentally observed [75]. The standard deviation linearly increases with the mean value, which defies the central limit theorem. The spatial variations in the axis of the order parameter field produce curvature-induced currents, which underlie the giant fluctuation. However, the theoretical analysis is complicated because of the anisotropic nature of binary collisions between rods.

Subsequently, a shape-isotropic active “polar disk” was designed and manufactured, featuring a special design that enables particles to move coherently along a direction strongly associated with their inherent polarity, resulting in a weak average effective alignment through intraparticle collisions. This system exhibits collective motion with orientationally-ordered regions and giant number fluctuations [76, 77]. Subsequently, they proposed a theoretical model that incorporates the intrinsic polarity of particles, see the Eq. (16), predicting the possibility of true long-range order in the experimental system [78].(16)

Here, φi presents the polarity angle with ni=(cos φi, sin φi), ξ describes the strength of the coupling between polarity and velocity, and is the angle between velocity and polarity.

To deepen our comprehension of nonequilibrium thermodynamics, it is imperative to re-evaluate numerous pivotal physical concepts and relationships within the context of an active thermal bath. It has been found that many physical laws that have long been well-established in a thermal bath in equilibrium state are no longer applicable to an active bath and require redefinition and generalization, with pressure being a typical example. According to equilibrium thermodynamics, pressure is a state quantity, which implies that pressure solely depends on the bulk properties of the system. However, pressure in the active bath is no longer a state function. A series of studies on pressure have been carried out from theoretical, numerical, and colloidal experimental approaches. Various methods have been proposed to determine the pressure of active systems across different dimensions and active systems, yet the fundamental form of the resulting pressure remains consistent [79-84]. Sandoval [85] even derived the theoretical expression for the pressure of active particles with inertia.(17)Equation (17) gives the pressure of interacting active particles, where F is an aligning torque between the particles, and the distribution depends on the wall potential [80].

Macroscopic, artificial active particles are also valuable for the exploration of pressure. The mechanical pressure exerted by 3D printed self-propelled polar disks onto various flexible unidimensional membranes is examined [86]. When the particles are present on both sides of the membrane, membrane instability are observed in experiments, aligning with the theoretical prediction, as shown in Figure 7. This suggests that pressure in the active bath is strongly influenced by the membrane, thus not being a state variable. Interestingly, by introducing the concept of “active impulse” or the conservation of momentum between active particles and background particles, the state equation of active pressure can be restored [87, 88]. Even more fascinating, after the collision of an active gas formed by Hexbug robots with an elastic membrane, two features that the dissipation rate of the membrane and the effective temperature of the active gas characterize the action of particles on the membrane, can be described by a Langevin equation with white noise and frequency-independent dissipation [89].

thumbnail Figure 7

Mechanical equilibrium and instability. (A) Mechanical equilibrium between a small number of self-propelled disks (on the right) and a large number of isotropic passive disks (on the left). (B) S-shaped instability observed when self-propelled disks are distributed in equal numbers on both sides of the membrane (Adapted with permission from Ref. [86]. Copyright c 2017, American Physical Society).

The velocity distribution of active particles is another attractive topic. The theory of homogeneously driven granular gases of hard particles predicts that the steady state is characterized by a velocity distribution function with overpopulated high-energy tails. By active granular rotors that overcome the limitations of previous experiments, the study demonstrates a high-energy tail distribution in the velocity distribution and its exponent is consistent with theoretical prediction, as shown in Figure 8B(b) [90].

thumbnail Figure 8

(A) (a) An active particle with seven legs manufactured by rapid prototyping; (b) top view of the system with N=379 particles. (B) (a) Distribution of the rotational velocity of asymmetric Gaussian shape (solid lines); (b) velocity distribution function P(|v|). The dashed lines show the function predicted by kinetic theory of granular gases, P(v)~e-3/2 (Adapted with permission from Ref. [90]. Copyright c 2017, American Physical Society).

Coincidentally, the velocity components measured by airflow-driven active rotors exhibit a Gaussian distribution, and the speed follows a Maxwell-Boltzmann distribution, suggesting that the driving type plays a key role in the statistical mechanics of granular gases [91]. However, Farhadi et al. [92] used a similar experimental system and found that the distribution of translational velocities in a single-particle system follows a Gaussian distribution, but when measured in a multi-particle system, it significantly diverges from the Gaussian distribution. This divergence is likely due to the fact that the spinners in their study rotated at such high speeds that collisions between rapidly spinning spinners generated translational velocities that contributed to significant deviations from the expected distribution. Beyond the statistical analysis of active particles’ translational velocities, a granular gas with only rotational degrees of freedom was also considered. In the experiment, each macro-scale rotor was confined to a circular hole in the lattice, thereby suppressing its translational degree of freedom. By meticulously analyzing the probability density function of angular velocity and its first and second moments, researchers revealed that the lattice constant had a dominant influence on the dynamics of the system rather than the topological structure of the lattice [93]. Collective effects are almost completely suppressed in the system consisting of multi particles of pure rotators, and thus the dynamics of the rotators are not effected by their collective motion.

In addition to the velocity statistics, the active bath has also been applied to the study of the noise. Cheng et al. [94] conducted an experimental study on the active noise experienced by a spring-constrained passive rotor immersed in an active bath. Through an analysis of the power spectrum of the rotor trajectory, they found that the active noise experienced by the rotor demonstrates a significant exponential time dependence, and the probability density function of this noise exhibits non-Gaussian characteristics. These experimental results provide direct experimental evidence in support of the Ornstein-Uhlenbeck model in active systems. Additionally, a monolayer of air-fluidized star-shaped particles, forming an active granular glass, has been used to detect Gardner crossover experimental signals. Xiao et al. [95] regulate the pressure by adjusting the tension applied to the closed boundary of the active quasi-thermal granular glass, and detect the Gardner crossover through measurements of cage size and separation order parameter of the particle position and orientation. The active particle bath has also been designed to test Markovianity in non-equilibrium systems [96]. In this experiment, a foam ball, serving as a probe particle, was placed in a parabolic bowl, also known as a two-dimensional harmonic potential well, along with an active bath of Hexbugs. Additionally, an external aerodynamic force from a fan was applied to the foam ball. The position probability density function of the random motion of the foam ball was measured. When the toy robots with two speed states were used, it was observed that the experimental violation of the nonlinear generalized fluctuation-dissipation relation occurred due to the introduction of an additional time scale into the process.(18)From the one-time probability density given by Eq. (18), the two steady-state distributions Pε(x) and P0(x), when , can be obtained [88].

Diffusion is another area of interest in the statistical physics of active matters. In a quasi-two-dimensional monolayer of air-driven granular rods, the anisotropy in particle shape leads to dynamic anisotropy and super-diffusive dynamics parallel to the long axis of the rod [97]. In this study, the active ornstein-uhlenbeck particles (AOUPs) model was employed to comprehend the diffusional dynamics by incorporating self-propulsion velocity into active noise with memory effects, thereby decoupling rotational and translational degrees of freedom. Similarly, the effect of packing fraction on the diffusion of self-propelled granular chains has been investigated. By measuring mean square displacement at different packing fractions, normal diffusion, caging effect and resulting scaling laws were observed [98]. If the vibrating substrate is replaced by a rough one, the rolling of short chains can significantly influence their dynamics due to the low moment of inertia of the chains and enhanced friction [99]. Rich and complex chiral diffusive behaviors, particularly odd diffusion, arise in active rotors. Odd diffusion is manifest as a two-dimensional tensor with an antisymmetric component. The breaking of time and parity reversal symmetry in the active rotors suggests the emergence of particle velocity cross-correlations, which necessarily leads to an asymmetric diffusion tensor T [100].(19)(20)With εij being the 2D Levi-Civita symbol, D(Dodd) represents the Brownian diffusion coefficient (odd diffusion coefficient).

Swarming, flocking and vortex

Swarming, flocking, and vortex are all manifestations of collection behaviors. While these phenomena typically occur in biological systems such as bird flocks, fish schools, and ant colonies, artificial active matter can also display similar behaviors. Previous research has examined the impact of vibrating rod shape on the nature of the orientational ordering. Particles with different shapes show tetratic, nematic, and smectic phases, particularly a defect-ridden nematic state characterized by large-scale swirling motion [101]. Specifically, cylindrical particles display prominent tetratic and nematic phases at high packing fractions. Those with thinner ends, resembling rolling pins or bails, feature a unique nematic state abundant in defects, resulting in large-scale swirling motion. Conversely, long-grain basmati rice, with a geometry between the two aforementioned shapes, demonstrates a distinct smectic order. Furthermore, it has been observed that self-propelled rods with sufficient activity engage in a continuous swirling motion, where velocity is closely linked to particle orientation [102]. The absence of clustering in round self-propelled particles underscores the critical role of particle shape in aggregation and collective dynamics. It is evident that particle shape significantly influences aggregation. Rod-shaped particles, for instance, show a considerably lower probability of turning around and departing from the wall compared to spherical particles, leading to their prolonged entrapment near the wall. Moreover, the collective motion of rods within the container is often obscured by intense random fluctuations, particularly at high driving accelerations of the container.

When millimeter-sized tapered rods are immersed in a medium containing spherical beads, a fascinating phase transition occurs as the rods transition from a state of disorder to one where their orientation vectors align with the increasing packing fraction of spherical beads. This transition is accompanied by a spontaneous alignment of both velocities and orientations, resulting in a remarkable degree of orientational order among the rods. At low densities of spherical beads, the rods move in random directions, while at higher densities, they move coherently. Remarkably, even at extremely low concentrations of rods, these rod-shaped particles exhibit the capacity for self-organization, transitioning into a globally oriented state of movement and spontaneously engaging in flocking behavior. The orientational information is transmitted across large distances via the flow of the ambient medium, which consists of millimeter-sized spherical beads. Consequently, an increase in the number of these spherical beads not only enhances the flocking behavior of the rod-shaped particles but also facilitates the coherent transport of the spherical beads themselves. The system undergoes a flocking transition, which depends on two crucial factors: moving rods dragging the beads and neighboring rods reorienting within the resultant flow (the weathercock effect) [103].

Similarly, granular rotors driven by airflow also exhibit a global vortex state, and interestingly, the rotation direction of the vortex is not necessarily the same as that of the rotors [104]. The identical chiral particle can generate flows with diverse chiralities. As the kinetic energy input increases, three vortex states emerge with different chiral modes: spinwise chirality, counter-spinwise chirality, and complex chirality. In the spinwise chirality mode, fluid rotation aligns with the particle’s spinwise direction. Conversely, in the counter-spinwise chirality mode, fluid rotation opposes the particle’s spinwise direction. The complex chirality mode features multiple vortices with different senses of rotation. Moreover, the transitions among these states are strongly influenced by the specific state of the statistical correlation between particle-spin and translational velocity.

Other collective behaviors

Spontaneous oscillation

As mentioned in Section “Swarming, flocking and vortex”, the spontaneous collective motions generally exhibit either chaotic flows like swarming, flocking and turbulence, or directed flows like active vortex. In contrast, experimental model systems designed to investigate the spontaneous collective oscillation of active particles are relatively scarce, despite their widespread occurrence in nonlinear and out-of-equilibrium systems, ranging from biological, optical, mechanical engineering systems to electric circuits. A recent report has described a spontaneous number oscillation observed in vibration-activated granular particles shuttling between two vessels connected by a narrow channel, as shown in Figure 9 [105]. A simple model based on the force balance and particle conservation can properly reproduce the experimental results without fitting parameters. Beyond spontaneous population oscillation, that of mechanical structures due to the interplay between activity and elasticity is another frontier in scientific community. By connecting hexbugs end-to-end, the resulting structure resembles a one-dimensional elastic chain. Under various coupling conditions, this active chain exhibits flagellar motion, self-snapping and synchronization [106]. By simple models of coupled pendula with follower forces, these transitions can be quantitatively clarified.

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A complete spontaneous oscillation process is displayed [105].

When compared to one-dimensional chains, two-dimensional active solids with more degrees of freedom possess more abundant and intriguing collective vibrational modes [107]. The active solid is constructed from a two-dimensional elastic lattice network. Each site (node) contains one hexbug, which is confined within the node but has a fluctuating orientation. The active solid combines the characteristics of elastic solids and active liquids. For sufficiently strong coupling, lattice nodes exhibit collective oscillating modes known as chiral phases near their equilibrium positions. The chiral phase originates from the single-particle dynamics, but the selection of modes is determined by nonlinear elasto-active feedback. Subsequent work has further confirmed that two collective behaviors, synchronous chiral oscillation, and global alternating rotation, can occur in the same active solid, and demonstrate how tension controls the transition between these collective behaviors [108].

Ratchet effects

Due to the spatiotemporal symmetry break, passive objects with asymmetric structures can extract energy from an active bath to perform directed translational or rotational motion. A simulation has shown that a mobile asymmetric gear immersed in a bacterial bath where motile bacteria were modeled as active rods with run-and-tumble dynamics, spontaneously rotates in a unidirectional manner [109]. Following the prediction, researchers have achieved the unidirectional rotation of micro-nano gears with asymmetric shapes using colloidal particles and bacteria [110, 111]. The same idea has also been explored in macroscopic systems. An asymmetric gear can rectify the random motion of Hexbugs in its vicinity, and in turn, the active robots can apply torque to the gear, driving it in a unidirectional rotation, see Figure 10. The rectification efficiency increases with the degree of gear asymmetry [112]. Later, the ratchet experiment is revisited by active short chains and similar results were obtained [113]. The geometric shape of the gear in the ratchet effect is usually non-deformable, meaning that the spatial symmetry break is predetermined. Li et al. [114] designed and manufactured geometrically deformable gears that undergo spontaneous deformation by the collision of macroscopic active particles and maintain the broken shape, yielding unidirectional rotation. It has been found that persistence time, self-propelled force and rotating friction are responsible for maintaining the broken symmetry and thus the rotation.

thumbnail Figure 10

Experimental setup. (A) Twenty robots move in an arena with a curved boundary. A gear at the center of the arena is free to rotate (around point O) but not to translate. Instantaneous robot translational velocities are by blue arrows. (B) Four gears with different geometric parameters used in the experiments (Adapted with permission from Ref. [112]. Copyright c 2013, EPLA).

Self-assembly

The phase separation of micro/mesoscopic active matter based on their motion properties can also be reproduced by macroscopic active agents. The collective motion of 3D printed rotors on a vibrating table showed that super-diffusive interfacial motion and phase separation between two kinds of chiral rotors occur through spinodal decomposition. Over longtime scales, spatial confinement is more favorable for symmetric separation [115]. These 3D printed rotors are also used to mimic surfactants [116]. Vibrot surfactants are chains formed by connecting two opposite chiral rotors through a rigid rod, as shown in the Figure 11. Due to the activity and chain geometry, double-stranded vibrot surfactants cannot form micelles but act as mixing ingredients. Single-stranded vibrot chains form spinning vesicles termed rotelles that can be constituted self-spinning particles of a higher degree of complexity. Chiral particles also exhibit stereoselectivity and chiral self-recognition in self-organization [117]. Without changing the shape or size, 3D printed granular ellipsoids can self-assemble into two types of particles depending on the chirality of the pairing ones. Upon measuring the lifespan of self-assembled particles, it was discovered that there exists a preferential particle pairing pattern within assemblies composed of ellipsoids with varying chiral activities.

thumbnail Figure 11

Double-stranded vibrot chains have an affinity for chiral vibrot fluid interfaces. (A) Double-stranded vibrot chain of geometry 2×6. (B) Experimental snapshots with a 2×14 chain at three observation times. Orange lines are connectors. (C) Evolution of the distance between the centers of mass of the left-handed and right-handed fraction of vibrots (orange curve), distance of each fraction from the chain’s center of mass (purple and green). (From Ref. [116]. Copyright c 2021, The Authors, some rights reserved; exclusive licensee AAAS. Distributed under a CC BY-NC 4.0 license. Reprinted with permission from AAAS).

Topological edge flow

When discussing collective behaviors of macroscopic active agents, which are usually present in a confined environment due to their size, the interactions between active particles and boundaries, both hydrodynamic and steric, become pivotal. A collection of active rotors made from Hexbugs, when placed in a confined container, demonstrates a spontaneous unidirectional boundary flow. This flow, viewed from a topological protection standpoint, is nonexistent without the boundary and remains robust across varying rotor densities and boundary shapes [118]. The robust boundary flow can be elucidated by mapping it onto the classical Su-Schrieffer-Heeger model derived from hydrodynamic theory. While topological boundary flows are typically discussed within the framework of incompressible and structureless homogeneous fluids, Liu et al. [119] observed an oscillating angular velocity along the radial direction in the context of vibrating granular rotors. To explain this anomalous behavior, a hydrodynamic theory incorporating compressibility and density inhomogeneity has been proposed. Specifically, density inhomogeneity gives rise to an oscillating rotational viscosity coefficient, which in turn impacts the position-dependent friction stress. The coupling of the rotor’s spin with the friction stress ultimately leads to oscillating motion. In subsequent work, they employed the same experimental system alongside numerical simulation to verify that the boundary flow can topologically transport materials, as shown in Figure 12 [120]. The key to this controllable transport lies in the odd viscosity, which can be modulated by rotor activity, thereby enhancing the effective attraction between the transported cargo and the boundary.

thumbnail Figure 12

(A) Top view (top) and side view (bottom) of the gearlike spinner in experiments. The arrow denotes the clockwise rotation of the spinner. (B) Sketch of the spinner in simulations. (C) Experimental and (D) simulation snapshots of a large passive disk transported in the chiral active fluid confined by a circular boundary with a rodlike obstacle. The arrows represent the direction of the edge transport (edge flow) [120].

The oscillating edge flow mentioned above can be obtained by solving the following equation [119, 120]:(21)the radial stress on the cargo can be expressed as(22)which shows the odd viscosity enhances the spinning-dependent radial stress due to η0vt(R)≥0.

Phase transition

Soni et al. [121] conducted a study on the phase transitions in a vibrating mixture consisting of active granular rods and passive beads at various packing fractions. The transitions observed included the transition from the isotropic phase to the homogeneous flocking phase, as well as the ordered band phase in between. Within the ordered band phase, they observed broken-symmetry sound modes and huge number fluctuations.

ROBOTS IN ACTIVE MATTER

Simple robots-Hexbugs

In vibration-activated matter, such as vibrating rods or 3D printed rotors, all individuals are simultaneously driven by the same excitation source. Compared to vibrating particles, robotic systems offer greater flexibility in control, as the motion of each individual robot can be independently controlled. As the simplest robot, HexBug, or bristle-bots (as illustrated in Figure 4 and 图13) is widely utilized in collective behaviors, attributed to its straightforward structure and budget-friendly price. The Hexbug is equipped with a battery-driven eccentric wheel motor, which generates vertical vibrations that are transmitted to rubber legs on each side of the robot’s body. The friction between the legs and the vibrating table can convert the vertical vibration into the horizontal directional motion for the Hexbug. Without any modifications to the interaction between Hexbugs, they transfer interactions through inelastic collisions. It is worth pointing out that, despite some literature’s classification of the Hexbugs as robots, they are overly simplistic, particularly lacking programmable chips, which prohibits effective information communication between individuals. Therefore, the Hexbugs cannot be accurately categorized as intelligent robots.

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Snapshot of the system during a typical temporal clogging. Red arrows indicate the feedback path followed by the Hexbugs (Adapted with permission from Ref. [123]. Copyright c 2017, American Physical Society).

In preceding sections, we have presented some intriguing work involving the utilization of Hexbugs, as exemplified in Refs. [60, 63-65, 67, 70, 71, 89, 96, 106, 107, 112, 118]. Now, we delve deeper into other important work. It has been discovered that when Hexbugs are confined within an arena with a soft boundary, the increase in the density promotes a transition from disordered and uncoordinated movements to organized collective motion, such as vortexes or collective stagnation. Both experimental and simulation results have confirmed that this transition is determined by the relative magnitude of the translation and rotation of the particle [122]. When confined within a hard circular boundary, Hexbugs with a high packing fraction form polar clusters near the boundary. While inside the arena, particles show gaslike behavior. The coexistence between gas state and boundary clusters is due to inertial effects. In contrast, the coupling of soft, movable boundaries and particle activity can regulate a wide range of interesting dynamics [38]. For instance, an artificial vesicle composed of Hexbugs and a flexible boundary can pass through a narrow hole or rotate around an obstacle.

Hexbugs can also serve as an alternative for living organisms in exploring the transition between flowing and clogging of discrete matter passing through a bottleneck, see Figure 13. The power-law characteristics of the tail of the complementary cumulative distribution function, observed in animal or human experiments, can be accurately reproduced in experiments utilizing Hexbugs. Increasing the number of particles in front of an exit can trigger a shift from a flowing to a clogging state. This clogging-unclogging transition is caused by a rapid surge in pressure near the exit, reminiscent of the “fast is slow” effect in crowd evacuation through a bottleneck [123]. Similarly, groups of Hexbugs moving through various environments can serve as models for traffic flow. When a collection of Hexbugs move unidirectionally through a ring channel, and the channel width gradually narrows, a transition from a flowing to a clogging state occurs. The coexistence region depends on fluctuations such as the distribution of the escape times from a clogging state [124].

Given the above studies, including those in previous sections, it is evident that Hexbug modifications and upgrades are highly flexible and convenient. For instance, it is feasible to increase the eccentric weight to convert a self-propelled Hexbug into a chiral one, assemble different shells to alter their geometric shapes, and even attach infrared or visible light sensors to manipulate their interactions with the environment. However, the absence of information exchange capabilities among individual Hexbugs, particularly programmable ones, restricts their potential to exhibit intricate collective behaviors and accomplish sophisticated tasks. Programmable robots effectively overcome this limitation, as they are equipped with communication chips designed to seamlessly receive and process information, enabling programmable interactions for enhanced functionality. In fact, programmable function introduces a new dimension of freedom in robots, enabling interactions that are not typically observed in most artificial active agents. As a result, a collection of robots demonstrates a richer and more diverse range of collective behaviors, embodying a “group intelligence” that enables them to accomplish tasks that are beyond the capabilities of individual robots. Moreover, they serve as a bionic platform for the evolution of ecosystems.

Intelligent robots

In contrast to active particles driven by vibration or simple robots limited to collision interactions, intelligent robots are equipped with multiple types of sensors that enable them to respond to light, electric, magnetic, and force fields, as well as to transmit information remotely between individuals. The multiple fields and multi-body coupling greatly enhance their functionality. For instance, an intricate cognitive ability, through locally interacting with each other in a specific setting, can emerge from a robot swarm, and the collective swarm can sequence tasks with a priori unknown execution order [125]. These robots even possess the ability to transform their surroundings, enabling a wider range of interactions with the environment and their peers.

The intelligent robotics system has the following advantages [126]: (1) Each robot exhibits independent motion. (2) Robots can effectively respond to various stimuli in their surrounding environments. (3) The interaction protocols between robots and between robots and the environments are relatively straightforward and programmable. (4) The system has no centralized control, and an individual robot has no access to global information. (5) Robots spontaneously coordinate to form a swarm, collaborating to accomplish specific tasks. These advantages impart significant robustness, scalability, and functional diversity to the collective behavior of swarm robots.

The swarm of robots involves the design, manufacture, and deployment of intelligent robots. As mentioned earlier, there are two research ideas on swarm robotics: one aims to continuously integrate advanced sensors, develop more sophisticated algorithms, introduce more complex forms of information interaction, achieve more intricate formations, and execute more difficult and flexible tasks. This approach gains significant attention from the engineering community. Alternatively, another research approach aims to minimize complexity and focuses solely on the most critical experimental parameters. This approach facilitates controlled experiments on theories of collective behaviors of active matter, enabling the discovery of novel phenomena, statistical laws, and underlying mechanisms. Through streamlined experimentation and precise control, new insights into the collective behavior of active matter can be gained, paving the way for future research and applications. The primary focus of this review is on the progress associated with the latter perspective. References to other achievements can be found in the accompanying review articles [127-129].

Through programmable local interaction and sensing capabilities, the 1024 Kilobot robots are capable of flexible self-assembly into a given two-dimensional shape, as shown in Figure 14 [130]. It is important to note that each robot is equipped with a predefined program that incorporates a fixed self-assembly algorithm and an image representing the target shape and size. Nevertheless, the emergence of this shape mentioned above is not entirely spontaneous, as each robot must have prior knowledge of the final shape, as well as its relative position within the group. Slavkov and his colleagues [131], inspired by the development of multicellular tissue morphology, solved the problem in the self-organization of a shape. Drawing upon the theory of reaction-diffusion circuits, especially the Turing system, they employed hundreds of Kilobot robots to achieve fully self-organized pattern without any robot self-positioning. Instead, they relied solely on local interactions with neighboring robots.

thumbnail Figure 14

Kilobot swarm robot. (L) A Kilobot robot, shown alongside a US penny for scale. (M) Each Kilobot has an onboard microcontroller for executing programs autonomously, two vibration motors for moving straight or turning on a flat surface, and a downward-facing infrared transmitter and receiver for the communication with others. Self-assembly experiments use up to 1024 physical robots. (A), (C), and (E) Desired shape provided to robots as part of their program. (B), (D) Self-assembly from initial starting positions of robots (left) to final self-assembled shape (right). (F) Completed assembly showing global warping of the shape due to individual robot errors. (G) Accuracy of shape formation is measured by comparing the true positions of each robot (red) and each robot’s internal localized position (gray). H–K Close-up images of starting seed robots (Reprinted with permission from Ref. [130], Copyright c 2016, AAAS).

To avoid undesired collisions between robots in a dense swarm, researchers have equipped them with exoskeletons. These exoskeletons not only enhance the robots’ mobility and stability in frequent collision scenarios, but also enable them to better respond to external fields and collisions [37]. Collisions are no longer considered a disadvantageous factor; rather, they facilitate information flow and online distributed learning, ultimately promoting collective performance. In contrast to the minimalist robot system mentioned below, the design of these robots above is still complex. Complex designs require robots to control individual components explicitly to perform specific functions. The malfunction of any one component often results in the failure of the entire robot. Li et al. [132] designed and manufactured a very simple robot that cannot move independently and has only one degree of freedom, limited to extending and contracting in the radial direction. Besides, the robot does not have individual identity or addressable position. A gradient algorithm embedded in robots and weak coupling via hanging magnets between individuals allow the robots to collectively move towards the light source and transport cargo, even when 20 percent of the robots malfunction. Due to the inherent scalability and robustness of this simple and seemingly-stochastic system, it provides an alternative approach to using sophisticated robots swarm for carrying out complex tasks. Furthermore, it has been found that the diversity of individual properties, despite often being considered inadequacy or disorder, can in fact offer another way alternative meaning of completing complex tasks [133]. The study provides experimental evidence that the intricate interplay between alignment and polydispersity of angular velocities in optically driven robots leads to self-sorting under confinement and enhancement of translational diffusion.

The intelligent robot swarm can also serve as a bio-inspired platform, enabling the dynamic coupling of individuals and their environment to be defined, quantified, and programmable. Wang et al. [134] have experimentally designed a novel type of intelligent robot based on the Kilobot system. Each robot is considered as a resource-driven entity that behaves like a living organism, consuming the resources (in this case, light) at its current location and moving towards a new position along the steepest local light intensity gradient. The resource field has a recovery rate, and the shadow area left behind by robots that have depleted resources can influence the behavior of other robots, ultimately creating a dynamically interactive and environmentally coupled active system. When the robot swarm resides within a continuously shrinking circular resource field, adjusting parameters such as the environmental recovery rate and the robot’s resource consumption rate allow the swarm to exhibit states comparable to those of material phases, including gaseous, solid, liquid, glassy, and jammed states, as well as dynamic transitions between these states. This research unveils hidden complexities and previously unexplored phenomena within ecosystems, providing a novel avenue for demonstrating swarm emergence behavior in the laboratory. Followed by the work above, they upgraded the robots by incorporating color sensitivity. Some robots prefer red light, while others favor blue light. These color preferences can be passed on between robots through information replication. Thus, the robots are endowed with mechanisms that mimic genetic, mutation, and natural selection, effectively importing the most critical ingredients of biological evolution into the inanimate robot collective, see Figure 15. By combining with dynamically programmable optical environments, robots can perceive and modify their surroundings, leading to intricate adaptive behaviors within the robot swarm [135]. This study employed a physical robotic system instead of computer simulations to investigate the characteristics of biological evolutionary systems, and arrived at conclusions that contrast with those from highly idealized computer simulations. In the same system, it was observed that during the environmental compression-expansion cycle, the robot swarm underwent a series of phase transitions related to boundaries. Additionally, there was a delay in the emergence of a time-reversal symmetry-breaking orderly hysteresis throughout the evolutionary process [136].

thumbnail Figure 15

(A) Depiction of the complex interactions of the robots. Ribbon: The robots move via self-generated field drive on a resource landscape generated by an underlying LED light board, which the robots sense. Field drive is the self-generated movement of the robots in response to local resource depletion. (B) The LED light board with robots across an RGB nested set of landscapes (Adapted with permission from Ref. [135]. Copyright c 2022, PNAS).

When multiple robots engage in collaborative motion, limited communication bandwidth can pose a notable challenge, as it introduces global communication delays. Surprisingly, the delay can in fact facilitate the emergence of coherent collective behavior among the robots [137]. Theoretically, an expression has been derived for the translational velocity of the robot swarm as a function of both time delay and delay coupling coefficient. The stable coherent modes predicted theoretically, including translation, ring state, and rotation, have been experimentally confirmed using two-wheeled robot vehicles. Notably, these modes can only be achieved through delay coupling. Beyond robot systems traversing on the ground, the collective formation and obstacle avoidance capabilities of flying robots have also undergone rigorous examination. Researchers have proposed a swarm model specifically tailored for real drones. This model seamlessly integrates an evolutionary optimization framework with carefully selected order parameters and fitness functions. To validate the effectiveness of this theoretical model, experiments have been conducted using a swarm of drones. These experiments have demonstrated fully autonomous and synchronized outdoor flight, as well as impressive obstacle avoidance within confined spaces [138].

Robots find applications in the construction of mechanical metamaterials as well. By connecting robots with flexible materials to form a one-dimensional structure, and implementing local control loops to break the reciprocity of the robot interactions, it becomes possible to generate spatial asymmetric standing waves and unidirectional amplified propagating waves across all frequencies. This unique capability can be harnessed to extract mechanical energy, effectively allowing energy to flow out from its source while preventing any backflow [139]. While there have been some in-depth studies on wave propagation in non-reciprocal media, the collective dynamics of multi-body systems in non-reciprocal interactions remains largely unexplored. Using bifurcation theory and non-Hermitian quantum mechanics, the emergence of these far-from-equilibrium phases is described. Three distinct self-organized non-reciprocal behaviors have been identified: synchronization, clustering, and pattern formation. In the context of synchronization, a carefully programmed robot swarm with nonreciprocal interaction was employed, successfully reproducing the phenomenon [140].

Unlike equilibrium states, where the emergence of order is associated with a decrease in free energy, in non-equilibrium systems, the occurrence of self-organization phenomena is typically preceded by a significant reduction in vibration amplitude. The “smarticle” intelligent robots were used to examine the changes in the angle of the three robot arms over time, serving as a verification of the principle of low rattling that underpins self-organized dynamical order [141]. Following the absorption of energy from the external environment, the system evolves towards a state of minimal energy expenditure—in other words, a state of low-amplitude rattling—thereby fostering the emergence of a steady state within the out-of-equilibrium system.

MACROSCOPIC PARTICLES IN A HYDRODYNAMIC ENVIRONMENT

In microscopic active particle systems, such as colloids and bacteria, the hydrodynamic environment is crucial. Not only does it provide the essential context for the survival of these agents, but it also significantly impacts the dynamics of the active particles. While most macroscopic and artificial active systems lack a hydrodynamic environment or the hydrodynamic environment can be disregarded, they are typically categorized as dry active systems. However, there exists a category of active particles within a hydrodynamic environment. Common examples include camphor boats, oil or pentanol droplets, and magnetic particles. Camphor boats and their variants are found moving at the liquid-air interface, while oil droplets and magnetic particles are typically situated within the liquid. For active particles such as camphor boats moving on the air-liquid interface, the Marangoni effect—caused by the concentration difference of camphor molecules in water—is responsible for directional motion. Theoretical models for the self-propelled mechanism can be found in Refs. [142-144].

The hydrodynamic environment not only propels the active motion of particles but also shapes the flow field around them, which in turn impacts the motion of other particles. In this way, hydrodynamics and activity coupling produce complex and rich individual and collective behaviors. The researchers discovered that a self-propelled camphor boat [145, 146] or an oil droplet [147-149] undergos spontaneous oscillations within channel-connected containers or fluidic channel, due to uneven surface tension. Subsequently, the oscillation of motion was also observed at the solid-liquid interface [150, 151].

In a hydrodynamic environment, a rapidly-rotating magnetic particle with a radius of 1 mm, generates a localized 3D vortex above a critical rotational frequency. This vortex and the magnetic particle stabilize into a quasiparticle known as the “spinner-vortex”, which demonstrates an effective attraction and dynamic trapping near the solid boundary of the container, enabling it to self-propel along any wall contour. The propulsion speed and the particle’s proximity to the boundary are determined by its angular velocity, which is balanced by the Magnus force and wall repulsion [152].

A variant of the camphor boat is the camphor ribbon—a rectangular paper strip saturated with camphor solution. Driven by the Marangoni force, this ribbon rotates on the air-water interface. When one end of the ribbon is secured with a pin, it spins around the pin. When two ribbons spin, coupled through the surrounding camphor layer, they exhibit two kinds of synchronization modes [153]. Interestingly, both the critical coupling coefficient and the initial condition are of utmost importance in estimating the synchronization region [154]. By introducing an additional camphor ribbon and arranging the three ribbons in a triangular or other configuration, a chimera state arises. In this state, one pair of ribbons rotates synchronously, while the remaining two pairs rotate uncoordinatedly [155, 156]. Furthermore, it has been experimentally observed that the ribbon undergoes a non-periodic abrupt rotation, characterized by irregularly alternating stationary and rotating states [157]. Synchronous rotation has been also observed between two such coupled non-periodic camphor rotors. One rotor follows the motion of the other rotor, due to occasional collisions between the coupled ribbons [158]. After the rotor is forced to stop and subsequently released, spontaneous reversal of the direction of rotation is observed. The probability of this reversal decreases with increasing stop time [159].

Just like the active chain assembled by linking Hexbugs, camphor disks connected by a string can also form an active chain. The chain exhibits spontaneous translational motion owing to the surface tension gradient. When one end of the chain is pivoted, regular oscillatory motion is generated. This mechanism may arise from the coupling between the orientation of disks on the chain and their instantaneous velocity [160]. A rotor with an asymmetrical shape floats on a liquid surface of a vertically vibrating bath, displaying bidirectional rotational motion in response to changes in the driving vibration frequency, as shown in Figure 16. The magnitude and direction of the angular velocity of rotation are controlled by the interaction between the rotor geometry and the wavelength of the fluid wave field generated by the rotor itself. Barotta et al. [161] have cleverly devised a chiral active particle with mass and geometric asymmetry, which can be remotely manipulated to move along the fluid interface by solely modulating the driving frequency.

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(A) Oblique perspective of a chiral spinner on the surface of a vibrating fluid bath. (B)–(D) The shadowgraph imaging technique was used to visualize the wave-field generated by the oscillating spinner and in (E)–(G) distributing tracer particles at the interface reveals qualitatively similar characteristic streaming-flow patterns for a typical spinner of size L=6.3 mm across the range of frequencies considered in experiments (Adapted with permission from Ref. [161]. Copyright c 2023, Springer Nature.).

Beyond the intriguing dynamics of an individual particle, noteworthy work has been undertaken to investigate collective behavior, which holds significant potential and importance. Camphor disks (called camphor boat), when placed on the surface of water within a circular container, generate repulsive forces among the particles. Once the particle count surpasses a specific threshold, they spontaneously transition from chaotic motion into an orderly hexagonal lattice configuration [162]. This emergent phenomenon is primarily attributed to the interplay among the inherent activity of the particles, Marangoni force, and the boundary confinement of the container. By modifying the physicochemical conditions, the camphor boats also displayed three distinct collective behaviors in the annular channel: a uniform state, congestion flow, and cluster flow [163]. Camphor boats are also used to understand the connection between active turbulence and classical turbulence. This approach offers a more accurate representation of the essence of active turbulence, as the flow field at the water surface is non-turbulent. The velocity statistics obtained from the camphor boats exhibit remarkable parallels between the active turbulence generated by these boats and classical turbulence theory. This similarity may serve as a valuable tool in gaining a deeper understanding of the origins and modeling of intermittency [159].

In a vibrating wave-driven matrix of vortices, resembling an optical lattice—also referred to as a fluid metamaterial—1 mm-sized ferromagnetic spinners are placed. These spinners are driven by a magnetic field, and by manipulating their spinning frequency and the vortex amplitude, a variety of collective behaviors can be observed: stable orbiting within the vortex, being trapped within the vortex, or escaping from the vortex cell and migrating to neighboring vortices [164]. In this way, these spinners serve as vehicles for transferring materials and information in the metamaterial. The interaction between the spinner and the flow within the liquid metamaterial creates a confinement for the particles. Thus, the manipulation of the Magnus force, which can counterbalance other radial forces in the flow, determines the radius of the orbit and the selection of the trapping unit can be precisely controlled. In parallel, Sáenz and colleagues introduced a hydrodynamic spin lattice consisting of an array of in-phase bouncing droplets. This active spin system serves as a macroscale analog of microscopic spin systems [165]. Their experiments showcased the bouncing behavior of active droplets on a vertically vibrating liquid surface. Under the activity-wave coupling, these droplets generate surface waves that gradually decay, leading the droplets to follow clockwise or counterclockwise circular trajectories and interact intricately. When hydrodynamic spin lattices are arranged on a one-dimensional or two-dimensional lattice with millimeter-scale spacing, the spin patterns exhibited by these droplets mimic the magnetic spin arrangements found in ferromagnetism or antiferromagnetism, depending on the lattice shape, size, and other experimental parameters.

OUTLOOK AND PERSPECTIVES

Due to its inherent non-equilibrium characteristics, macroscopic artificial active matter has given rise to a wealth of diverse phase states and numerous counterintuitive dynamical behaviors. It can serve as an ideal experimental platform, providing unprecedented opportunities for examining and advancing the non-equilibrium and soft matter physics. The field of the macroscopic artificial active matter encompasses mechanics, electrical engineering, computer science, and physics, and the integration of diverse elements has infused vitality into the study of active matter. In current and future research, it is one of the development trends to couple more degrees of freedom with active matter. As mentioned earlier, the importance of elasticity is increasingly being recognized in the study of active matter. Active particles, or their surrounding environment, have the ability to deform either actively or passively, and the elastic coupling of activity can lead to the emergence of novel and intriguing behaviors. For example, the interplay between activity and elasticity can cause collective oscillations, and a deformable active vesicle can pass through narrower constraint. Introducing the hydrodynamic effect into macroscopic artificial active matter is also a promising avenue of research. Active particles at the air-liquid interface or within the fluid can induce novel phenomena and manipulation that are not available in “dry” active matter, thanks to the surface tension and the long-range hydrodynamic interaction of flow fields. The integration of elasticity and hydrodynamic environments endows artificial active matter with a greater resemblance to biological systems, thereby enhancing their persuasiveness in the construction of bionic platforms and the analogue to real biological systems. Researchers have even begun exploring the use of elasticity, hydrodynamic effects, and activities in the construction of active metamaterials. In addition, shape can also produce unexpected behaviors or functions, and geometric coupling activity is also one of the directions that have not yet been fully explored. Furthermore, shape can yield unexpected behaviors or functions, making geometric coupling activity a promising field that remains largely unexplored. For example, active chains have the potential to establish analogies with polymer chains.

The information exchange between individuals in active matter is also one of the promising fields for future research. Currently, the majority of interactions among active particles belongs to passive responses, with only a limited number exhibiting active responses aimed at adapting to, or even transforming, the surrounding environment. As a typical representation of macroscopic artificial active matter, intelligent robots are well suited for manipulating information exchange and are intrinsically connected to machine learning and artificial intelligence. By training machine learning algorithms, the behavior of active particles can be predicted or optimized based on the environmental parameters provided as input (such as temperature, pH value, flow rate, etc.). This contributes to the establishment of a more advanced particle control system, thereby enhancing the responsiveness, as well as the precision of particle control. Using artificial intelligence technology, the behavior of active particles can also be customized according to specific needs. For example, through deep learning algorithms, it is possible to predict the trajectories and behavior patterns of particles with different shapes, materials, and sizes under different conditions, thereby enabling personalized particle design and application. Furthermore, by integrating sensors and navigation algorithms, macroscopic artificial active particles, including drones, can achieve autonomous navigation, formation, and obstacle avoidance capabilities. This enables particles to perform more efficient movements and execute tasks in complex environments. Thus, in addition to the potential applications in the biomedical field, active particles have opened up new application scenarios. Finally, through the design of novel information interaction principles, or even some principles that are not found in nature, it is possible to create structures and functions that do not exist in nature.

The macroscopic artificial active system possesses great advantages in science outreach as well. In contrast to microscopic or mesoscopic active matter, macroscopic artificial active particles, thanks to their large sizes, display intriguing motion patterns that are not only observable to the naked eye, but also significantly simpler for the general public to comprehend. Compared to microscopic experiments, the experimental operation of macroscopic particles is relatively straightforward. Additionally, the requisite equipment and materials are rather conventional, thereby facilitating convenient demonstration and exhibition. The investigation of macroscopic artificial active matter encompasses a multitude of disciplines, including physics, chemistry, and biology. This makes them an exemplary candidate for interdisciplinary education, effectively enhancing the comprehension of the interconnections among diverse scientific domains among the audience. The combined advantages of macroscopic artificial active particles render them an engaging and enlightening spotlight in science outreach events.

Acknowledgments

We appreciate M. C. Yang for helpful discussions.

Funding

This work was supported by the National Natural Science Foundation of China (12374205, 12304245 and 12364029), the Science Foundation of China University of Petroleum, Beijing (2462023YJRC031 and 2462024BJRC010), the Beijing Institute of Technology Research Fund Program for Young Scholars, the Young Elite Scientist Sponsorship Program by BAST (BYESS2023300), the Natural Science Foundation of Inner Mongolia Autonomous Region (2023QN01015), the Beijing National Laboratory for Condensed Matter Physics (2023BNLCMPKF014), the Academic Research Fund from the Singapore Ministry of Education Tier 1 Gant (RG59/21), and the National Research Foundation, Singapore, under its 29th Competitive Research Programme (CRP) Call (Award ID NRF-CRP29-2022-0002).

Author contributions

L.N., H.Z., J.Y. and Q.Z. collated and summarized the literature, and wrote the manuscript. N.Z., R.N. and P.L. guided the manuscript. All authors discussed and commented on the manuscript.

Conflict of interest

The authors declare no conflict of interest.

References

All Tables

Table 1

Table 1Examples of experimentally realized artificial active particles. The letters in the first column correspond to the examples plotted in Figure 1 [2]

In the text

All Figures

thumbnail Figure 1

Active particles are macroscopic, microscopic and nanoscopic in size and have propulsion speeds (typically) up to a fraction of a centimeter per second. The letters correspond to the artificial particles in Table 1 [16-39]. The insets show examples of biological and artificial swimmers. Three representative particles of macroscopic artificial active matter are displayed in the green box: robots, Hexbugs, and surfers (Adapted with permission from Ref. [2]. Copyrightc2016, American Physical Society).
In the text
thumbnail Figure 2

Mean square displacement of a macroscopic active particle with different self-propelled velocities. For a passive Brownian particle (v0=0 cm s-1), the motion is always normal diffusion [MSD ], while for an active particle, the motion is diffusive ballistic at intermediate time scales [MSD ], and again normal diffusion but with an enhanced diffusion coefficient at long time scales.
In the text
thumbnail Figure 3

Illustration for self-propelled mechanism of different particles (A) Chiral bent-wire (Adapted with permission from Ref. [52]. Copyright c 2005, American Physical Society). (B) Vibrating vibrot (Adapted with permission from Ref. [53]. Copyright c 2016, IOP Publishing Ltd. and Deutsche Physikalische Gesellschaft). (C) 3D printed walkers, in which structures in top row have connected (rigid) legs and structures in bottom row have free (flexible) legs (Adapted with permission from Ref. [51]. Copyright c 2016, IOP Publishing Ltd and Deutsche Physikalische Gesellschaft). (D) Self-propelled dimer made of two grain-filled balls. Upper panel, instantaneous horizontal velocity vs. time. Lower panel, vertical position of both balls and the vibrating plate as a function of time [57] (Adapted with permission from Ref. [57]. Copyright c 2017, Spring Nature).
In the text
thumbnail Figure 4

(A) Hexbug running in a parabolic dish; (B) close-up picture of a Hexbug. n denotes the particle orientation. (c) Observed orbiting trajectory, the arrows represent the particle orientation n and self-propelled velocity v. The color codes are representative of the passage of time, aiding us in distinguishing and identifying the trajectories of the Hexbug over time (Adapted with permission from Ref. [63]. Copyright c 2019, American Physical Society).
In the text
thumbnail Figure 5

(A) One vehicle transiting around a central depression; (B) side view of the differential driven vehicle; (C) different k(r) functions for prograde (light blue) and retrograde (dark blue) precession measured from experiments. The decreasing k has the same trend as the measured tilt angle γ(r) (black stars) (Adapted with permission from Ref. [66]. Copyright c 2022, PNAS).
In the text
thumbnail Figure 6

(A) (a) Top view of the sorting experiment with right-turning particles and left-turning particles; (b) particle extraction rate as a function of the total number of particles in the channel for the horizontal (α=0°) and the tilted experiments (α=1.8°); (c) particle locations in the channel without tilt; (d) particle locations in the channel with a tilt α=1.8° ((A) (a)–(d) Adapted from Ref. [70]. Copyright c 2020, American Physical Society). (B) Typical trapped and untrapped states in experiment and simulation (Adapted with permission from Ref. [71]. Copyright c 2019, American Physical Society).
In the text
thumbnail Figure 7

Mechanical equilibrium and instability. (A) Mechanical equilibrium between a small number of self-propelled disks (on the right) and a large number of isotropic passive disks (on the left). (B) S-shaped instability observed when self-propelled disks are distributed in equal numbers on both sides of the membrane (Adapted with permission from Ref. [86]. Copyright c 2017, American Physical Society).
In the text
thumbnail Figure 8

(A) (a) An active particle with seven legs manufactured by rapid prototyping; (b) top view of the system with N=379 particles. (B) (a) Distribution of the rotational velocity of asymmetric Gaussian shape (solid lines); (b) velocity distribution function P(|v|). The dashed lines show the function predicted by kinetic theory of granular gases, P(v)~e-3/2 (Adapted with permission from Ref. [90]. Copyright c 2017, American Physical Society).
In the text
thumbnail Figure 9

A complete spontaneous oscillation process is displayed [105].
In the text
thumbnail Figure 10

Experimental setup. (A) Twenty robots move in an arena with a curved boundary. A gear at the center of the arena is free to rotate (around point O) but not to translate. Instantaneous robot translational velocities are by blue arrows. (B) Four gears with different geometric parameters used in the experiments (Adapted with permission from Ref. [112]. Copyright c 2013, EPLA).
In the text
thumbnail Figure 11

Double-stranded vibrot chains have an affinity for chiral vibrot fluid interfaces. (A) Double-stranded vibrot chain of geometry 2×6. (B) Experimental snapshots with a 2×14 chain at three observation times. Orange lines are connectors. (C) Evolution of the distance between the centers of mass of the left-handed and right-handed fraction of vibrots (orange curve), distance of each fraction from the chain’s center of mass (purple and green). (From Ref. [116]. Copyright c 2021, The Authors, some rights reserved; exclusive licensee AAAS. Distributed under a CC BY-NC 4.0 license. Reprinted with permission from AAAS).
In the text
thumbnail Figure 12

(A) Top view (top) and side view (bottom) of the gearlike spinner in experiments. The arrow denotes the clockwise rotation of the spinner. (B) Sketch of the spinner in simulations. (C) Experimental and (D) simulation snapshots of a large passive disk transported in the chiral active fluid confined by a circular boundary with a rodlike obstacle. The arrows represent the direction of the edge transport (edge flow) [120].
In the text
thumbnail Figure 13

Snapshot of the system during a typical temporal clogging. Red arrows indicate the feedback path followed by the Hexbugs (Adapted with permission from Ref. [123]. Copyright c 2017, American Physical Society).
In the text
thumbnail Figure 14

Kilobot swarm robot. (L) A Kilobot robot, shown alongside a US penny for scale. (M) Each Kilobot has an onboard microcontroller for executing programs autonomously, two vibration motors for moving straight or turning on a flat surface, and a downward-facing infrared transmitter and receiver for the communication with others. Self-assembly experiments use up to 1024 physical robots. (A), (C), and (E) Desired shape provided to robots as part of their program. (B), (D) Self-assembly from initial starting positions of robots (left) to final self-assembled shape (right). (F) Completed assembly showing global warping of the shape due to individual robot errors. (G) Accuracy of shape formation is measured by comparing the true positions of each robot (red) and each robot’s internal localized position (gray). H–K Close-up images of starting seed robots (Reprinted with permission from Ref. [130], Copyright c 2016, AAAS).
In the text
thumbnail Figure 15

(A) Depiction of the complex interactions of the robots. Ribbon: The robots move via self-generated field drive on a resource landscape generated by an underlying LED light board, which the robots sense. Field drive is the self-generated movement of the robots in response to local resource depletion. (B) The LED light board with robots across an RGB nested set of landscapes (Adapted with permission from Ref. [135]. Copyright c 2022, PNAS).
In the text
thumbnail Figure 16

(A) Oblique perspective of a chiral spinner on the surface of a vibrating fluid bath. (B)–(D) The shadowgraph imaging technique was used to visualize the wave-field generated by the oscillating spinner and in (E)–(G) distributing tracer particles at the interface reveals qualitatively similar characteristic streaming-flow patterns for a typical spinner of size L=6.3 mm across the range of frequencies considered in experiments (Adapted with permission from Ref. [161]. Copyright c 2023, Springer Nature.).
In the text

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