Self-assembly of active colloidal molecules with dynamic function

Defect Solutions of the Nonreciprocal Cahn-Hilliard Model: Spirals and Targets

We study the defect solutions of the nonreciprocal Cahn-Hilliard model. We find two kinds of defects, spirals with unit magnitude topological charge, and topologically neutral targets. These defects generate radially outward traveling waves and thus break the parity and time-reversal symmetry. For a given strength of nonreciprocity, spirals and targets with unique asymptotic wave number and amplitude are selected. We use large-scale simulations to show that at low nonreciprocity α, disordered states evolve into quasistationary spiral networks. With increasing α, we observe networks composed primarily of targets. Beyond a critical threshold α_{c}, a disorder-order transition from defect networks to traveling waves emerges. The transition is marked by a sharp rise in the global polar order.

Impact of non-reciprocal interactions on colloidal self-assembly with tunable anisotropy

Non-reciprocal (NR) effective interactions violating Newton's third law occur in many biological systems, but can also be engineered in synthetic, colloidal systems. Recent research has shown that such NR interactions can have tremendous effects on the overall collective behavior and pattern formation, but can also influence aggregation processes on the particle scale. Here, we focus on the impact of non-reciprocity on the self-assembly of a colloidal system (originally passive) with anisotropic interactions whose character is tunable by external fields. In the absence of non-reciprocity, that is, under equilibrium conditions, the colloids form square-like and hexagonal aggregates with extremely long lifetimes yet no large-scale phase separation [Kogler et al., Soft Matter 11, 7356 (2015)], indicating kinetic trapping. Here, we study, based on Brownian dynamics simulations in 2D, an NR version of this model consisting of two species with reciprocal isotropic, but NR anisotropic interactions. We find that NR induces an effective propulsion of particle pairs and small aggregates ("active colloidal molecules") forming at the initial stages of self-assembly, an indication of the NR-induced non-equilibrium. The shape and stability of these initial clusters strongly depend on the degree of anisotropy. At longer times, we find, for weak NR interactions, large (even system-spanning) clusters where single particles can escape and enter at the boundaries, in stark contrast to the small rigid aggregates appearing at the same time in the passive case. In this sense, weak NR shortcuts the aggregation. Increasing the degree of NR (and thus, propulsion), we even observe large-scale phase separation if the interactions are weakly anisotropic. In contrast, systems with strong NR and anisotropy remain essentially disordered. Overall, the NR interactions are shown to destabilize the rigid aggregates interrupting self-assembly and phase separation in the passive case, thereby helping the system to overcome kinetic barriers.

Swarm Autonomy: From Agent Functionalization to Machine Intelligence

Swarm behaviors are common in nature, where individual organisms collaborate via perception, communication, and adaptation. Emulating these dynamics, large groups of active agents can self-organize through localized interactions, giving rise to complex swarm behaviors, which exhibit potential for applications across various domains. This review presents a comprehensive summary and perspective of synthetic swarms, to bridge the gap between the microscale individual agents and potential applications of synthetic swarms. It is begun by examining active agents, the fundamental units of synthetic swarms, to understand the origins of their motility and functionality in the presence of external stimuli. Then inter-agent communications and agent-environment communications that contribute to the swarm generation are summarized. Furthermore, the swarm behaviors reported to date and the emergence of machine intelligence within these behaviors are reviewed. Eventually, the applications enabled by distinct synthetic swarms are summarized. By discussing the emergent machine intelligence in swarm behaviors, insights are offered into the design and deployment of autonomous synthetic swarms for real-world applications.

Depletion-Induced Self-Assembly of Colloidal Particles on a Solid Substrate

We investigate the depletion contributions to the self-assembly of microcolloids on solid substrates. The assembly is driven by the exclusion of nanoparticles and nonadsorbing polymers from the depletion zone between the microcolloids in the liquid and the underlying substrate. The model system consists of 1 μm polystyrene particles that we deposit on a flat glass slab in an electrolyte solution. Using polystyrene nanoparticles and poly(acrylic acid) polymers as depleting agents, we demonstrate in our experiments that nanoparticle concentrations of 0.5% (w/v) support well-ordered packing of microcolloids on glass, while the presence of polymers leads to irregular aggregate deposition structures. A mixture of nanoparticles and polymers enhances the formation of colloidal aggregate and particulate surface coverage compared to using the polymers alone as a depletion agent. Moreover, tuning the polymer ionization state from pH 4 to 9 modifies the polymer conformational state and radius of gyration, which in turn alters the microcolloid deposition from compact multilayers to flocculated structures. Our study provides entropic strategies for manipulating particulate assembly on substrates from dispersed to continuous coatings.

Chemotactic particles as strong electrolytes: Debye-Hückel approximation and effective mobility law

We consider a binary mixture of chemically active particles that produce or consume solute molecules and that interact with each other through the long-range concentration fields they generate. We analytically calculate the effective phoretic mobility of these particles when the mixture is submitted to a constant, external concentration gradient, at leading order in the overall concentration. Relying on an analogy with the modeling of strong electrolytes, we show that the effective phoretic mobility decays with the square root of the concentration: our result is, therefore, a nonequilibrium counterpart to the celebrated Kohlrausch and Debye-Hückel-Onsager conductivity laws for electrolytes, which are extended here to particles with long-range nonreciprocal interactions. The effective mobility law we derive reveals the existence of a regime of maximal mobility and could find applications in the description of nanoscale transport phenomena in living cells.

Self-organization of active colloids mediated by chemical interactions

Self-propelled colloidal particles exhibit rich non-equilibrium phenomena and have promising applications in fields such as drug delivery and self-assembled active materials. Previous experimental and theoretical studies have shown that chemically active colloids that consume or produce a chemical can self-organize into clusters with diverse characteristics depending on the effective phoretic interactions. In this paper, we investigate self-organization in systems with multiple chemical species that undergo a network of reactions and multiple colloidal species that participate in different reactions. Active colloids propelled by complex chemical reactions with potentially nonlinear kinetics can be realized using enzymatic reactions that occur on the surface of enzyme-coated particles. To demonstrate how the self-organizing behavior depends on the chemical reactions active colloids catalyze and their chemical environment, we consider first a single type of colloid undergoing a simple catalytic reaction, and compare this often-studied case with self-organization in binary mixtures of colloids with sequential reactions, and binary mixtures with nonlinear autocatalytic reactions. Our results show that in general active colloids at low particle densities can form localized clusters in the presence of bulk chemical reactions and phoretic attractions. The characteristics of the clusters, however, depend on the reaction kinetics in the bulk and on the particles and phoretic coefficients. With one or two chemical species that only undergo surface reactions, the space for possible self-organizations are limited. By considering the additional system parameters that enter the chemical reaction network involving reactions on the colloids and in the fluid, the design space of colloidal self-organization can be enlarged, leading to a variety of non-equilibrium structures.

Enhanced diffusion of tracer particles in nonreciprocal mixtures

We study the diffusivity of a tagged particle in a binary mixture of Brownian particles with nonreciprocal interactions. Numerical simulations reveal that, for a broad class of interaction potentials, nonreciprocity can significantly increase the long-time diffusion coefficient of tracer particles and that this diffusion enhancement is associated with a breakdown of the Einstein relation. These observations are quantified and confirmed via two different and complementary analytical approaches: (i) a linearized stochastic density field theory, which is particularly accurate in the limit of soft interactions, and (ii) a reduced two-body description, which is exact at leading order in the density of particles. The latter reveals that diffusion enhancement can be attributed to the formation of transiently propelled dimers of particles, whose cohesion and speed are controlled by the nonreciprocal interactions.

Nonreciprocal interactions give rise to fast cilium synchronization in finite systems

Motile cilia beat in an asymmetric fashion in order to propel the surrounding fluid. When many cilia are located on a surface, their beating can synchronize such that their phases form metachronal waves. Here, we computationally study a model where each cilium is represented as a spherical particle, moving along a tilted trajectory with a position-dependent active driving force and a position-dependent internal drag coefficient. The model thus takes into account all the essential broken symmetries of the ciliary beat. We show that taking into account the near-field hydrodynamic interactions, the effective coupling between cilia even over an entire beating cycle can become nonreciprocal: The phase of a cilium is more strongly affected by an adjacent cilium on one side than by a cilium at the same distance in the opposite direction. As a result, synchronization starts from a seed at the edge of a group of cilia and propagates rapidly across the system, leading to a synchronization time that scales proportionally to the linear dimension of the system. We show that a ciliary carpet is characterized by three different velocities: the velocity of fluid transport, the phase velocity of metachronal waves, and the group velocity of order propagation. Unlike in systems with reciprocal coupling, boundary effects are not detrimental for synchronization, but rather enable the formation of the initial seed.

Self-organization of primitive metabolic cycles due to non-reciprocal interactions

One of the greatest mysteries concerning the origin of life is how it has emerged so quickly after the formation of the earth. In particular, it is not understood how metabolic cycles, which power the non-equilibrium activity of cells, have come into existence in the first instances. While it is generally expected that non-equilibrium conditions would have been necessary for the formation of primitive metabolic structures, the focus has so far been on externally imposed non-equilibrium conditions, such as temperature or proton gradients. Here, we propose an alternative paradigm in which naturally occurring non-reciprocal interactions between catalysts that can partner together in a cyclic reaction lead to their recruitment into self-organized functional structures. We uncover different classes of self-organized cycles that form through exponentially rapid coarsening processes, depending on the parity of the cycle and the nature of the interaction motifs, which are all generic but have readily tuneable features.

Active Colloids as Models, Materials, and Machines

Active colloids use energy input at the particle level to propel persistent motion and direct dynamic assemblies. We consider three types of colloids animated by chemical reactions, time-varying magnetic fields, and electric currents. For each type, we review the basic propulsion mechanisms at the particle level and discuss their consequences for collective behaviors in particle ensembles. These microscopic systems provide useful experimental models of nonequilibrium many-body physics in which dissipative currents break time-reversal symmetry. Freed from the constraints of thermodynamic equilibrium, active colloids assemble to form materials that move, reconfigure, heal, and adapt. Colloidal machines based on engineered particles and their assemblies provide a basis for mobile robots with increasing levels of autonomy. This review provides a conceptual framework for understanding and applying active colloids to create material systems that mimic the functions of living matter. We highlight opportunities for chemical engineers to contribute to this growing field.

Particle entity in the Doi-Peliti and response field formalisms

We introduce a procedure to test a theory for point particle entity, that is, whether said theory takes into account the discrete nature of the constituents of the system. We then identify the mechanism whereby particle entity is enforced in the context of two field-theoretic frameworks designed to incorporate the particle nature of the degrees of freedom, namely the Doi-Peliti field theory and the response field theory that derives from Dean's equation. While the Doi-Peliti field theory encodes the particle nature at a very fundamental level that is easily revealed, demonstrating the same for Dean's equation is more involved and results in a number of surprising diagrammatic identities. We derive those and discuss their implications. These results are particularly pertinent in the context of active matter, whose surprising and often counterintuitive phenomenology rests wholly on the particle nature of the agents and their degrees of freedom as particles.

Non-reciprocal multifarious self-organization

A hallmark of living systems is the ability to employ a common set of building blocks that can self-organize into a multitude of different structures. This capability can only be afforded in non-equilibrium conditions, as evident from the energy-consuming nature of the plethora of such dynamical processes. To achieve automated dynamical control of such self-assembled structures and transitions between them, we need to identify the fundamental aspects of non-equilibrium dynamics that can enable such processes. Here we identify programmable non-reciprocal interactions as a tool to achieve such functionalities. The design rule is composed of reciprocal interactions that lead to the equilibrium assembly of the different structures, through a process denoted as multifarious self-assembly, and non-reciprocal interactions that give rise to non-equilibrium dynamical transitions between the structures. The design of such self-organized shape-shifting structures can be implemented at different scales, from nucleic acids and peptides to proteins and colloids.

Interactions in active colloids

The past two decades have seen a remarkable progress in the development of synthetic colloidal agents which are capable of creating directed motion in an unbiased environment at the microscale. These self-propelling particles are often praised for their enormous potential to self-organize into dynamic nonequilibrium structures such as living clusters, synchronized super-rotor structures or self-propelling molecules featuring a complexity which is rarely found outside of the living world. However, the precise mechanisms underlying the formation and dynamics of many of these structures are still barely understood, which is likely to hinge on the gaps in our understanding of how active colloids interact. In particular, besides showing comparatively short-ranged interactions which are well known from passive colloids (Van der Waals, electrostatic etc), active colloids show novel hydrodynamic interactions as well as phoretic and substrate-mediated 'osmotic' cross-interactions which hinge on the action of the phoretic field gradients which are induced by the colloids on other colloids in the system. The present article discusses the complexity and the intriguing properties of these interactions which in general are long-ranged, non-instantaneous, non-pairwise and non-reciprocal and which may serve as key ingredients for the design of future nonequilibrium colloidal materials. Besides providing a brief overview on the state of the art of our understanding of these interactions a key aim of this review is to emphasize open key questions and corresponding open challenges.

Non-equilibrium phase separation in mixtures of catalytically active particles: size dispersity and screening effects

Biomolecular condensates in cells are often rich in catalytically active enzymes. This is particularly true in the case of the large enzymatic complexes known as metabolons, which contain different enzymes that participate in the same catalytic pathway. One possible explanation for this self-organization is the combination of the catalytic activity of the enzymes and a chemotactic response to gradients of their substrate, which leads to a substrate-mediated effective interaction between enzymes. These interactions constitute a purely non-equilibrium effect and show exotic features such as non-reciprocity. Here, we analytically study a model describing the phase separation of a mixture of such catalytically active particles. We show that a Michaelis-Menten-like dependence of the particles' activities manifests itself as a screening of the interactions, and that a mixture of two differently sized active species can exhibit phase separation with transient oscillations. We also derive a rich stability phase diagram for a mixture of two species with both concentration-dependent activity and size dispersity. This work highlights the variety of possible phase separation behaviours in mixtures of chemically active particles, which provides an alternative pathway to the passive interactions more commonly associated with phase separation in cells. Our results highlight non-equilibrium organizing principles that can be important for biologically relevant liquid-liquid phase separation.

Asymmetric acoustic wave scattering by a nonreciprocal and position-dependent mass defect

We investigate the asymmetric wave scattering in a phononic one-dimensional lattice with a nonreciprocal defect and position dependent masses coupled by the defect spring. The nonreciprocal interaction is characterized by a single parameter Δ while the nonlinear contribution due to position-dependent masses are controlled by a parameterχ. The transmission and reflection coefficients are analytically computed and the effects of the nonreciprocity and nonlinearity are detailed. We show that, in opposite with the linear case, the rectification factor has a frequency dependence, which leads to a more efficient diode-like action at large wavevectors. Further, the nonlinearity leads to an asymmetry of the reflected component, absent in the linear regime. We extend our analysis to a system with frictional forces which suppresses the multistability window promoted by the nonlinear mass contribution without compromising the rectification action.

Interconnectivity between Surface Reactivity and Self-Assembly of Kemp Elimination Catalyzing Nanorods

Understanding the fundamental facts behind dynamicity of catalytic processes has been a longstanding quest across disciplines. Herein, we report self-assembly of catalytically active gold nanorods that can be regulated by tuning its reactivity towards a proton transfer reaction at different pH. Unlike substrate-induced templating and co-operativity, the enhanced aggregation rate is due to alteration of catalytic surface charge only during reactivity as negatively charged transition state of reactant (5-nitrobenzisoxazole) is formed on positively charged nanorod while undergoing a concerted E2-pathway. Herein, enhanced diffusivity during catalytic processes might also act as an additional contributing factor. Furthermore, we have also shown that nanosized hydrophobic cavities of clustered nanorods can also efficiently accelerate the rate of an aromatic nucleophilic substitution reaction, which also demonstrates a catalytic phenomenon that can lead to cascading of other reactions where substrates and products of the starting reactions are not directly involved.

Revealing pseudorotation and ring-opening reactions in colloidal organic molecules

Colloids have a rich history of being used as 'big atoms' mimicking real atoms to study crystallization, gelation and the glass transition of condensed matter. Emulating the dynamics of molecules, however, has remained elusive. Recent advances in colloid chemistry allow patchy particles to be synthesized with accurate control over shape, functionality and coordination number. Here, we show that colloidal alkanes, specifically colloidal cyclopentane, assembled from tetrameric patchy particles by critical Casimir forces undergo the same chemical transformations as their atomic counterparts, allowing their dynamics to be studied in real time. We directly observe transitions between chair and twist conformations in colloidal cyclopentane, and we elucidate the interplay of bond bending strain and entropy in the molecular transition states and ring-opening reactions. These results open the door to investigate complex molecular kinetics and molecular reactions in the high-temperature classical limit, in which the colloidal analogue becomes a good model.

Regimes of motion of magnetocapillary swimmers

The dynamics of a triangular magnetocapillary swimmer is studied using the lattice Boltzmann method. We extend on our previous work, which deals with the self-assembly and a specific type of the swimmer motion characterized by the swimmer's maximum velocity centred around the particle's inverse viscous time. Here, we identify additional regimes of motion. First, modifying the ratio of surface tension and magnetic forces allows to study the swimmer propagation in the regime of significantly lower frequencies mainly defined by the strength of the magnetocapillary potential. Second, introducing a constant magnetic contribution in each of the particles in addition to their magnetic moment induced by external fields leads to another regime characterized by strong in-plane swimmer reorientations that resemble experimental observations.

Demixing of two species via reciprocally concentration-dependent diffusivity

We propose a model for demixing of two species by assuming a density-dependent effective diffusion coefficient of the particles. Both sorts of microswimmers diffuse as active overdamped Brownian particles with a noise intensity that is determined by the surrounding density of the respective other species within a sensing radius r_{s}. A higher concentration of the first (second) sort will enlarge the diffusion and, in consequence, the intensity of the noise experienced by the second (first) sort. Numerical and analytical investigations of steady states of the macroscopic equations prove the demixing of particles due to this reciprocally concentration-dependent diffusivity. An ambiguity of the numerical integration scheme for the purely local model (r_{s}→0) is resolved by considering nonvanishing sensing radii in a nonlocal model with r_{s}>0.

Non-equilibrium interaction between catalytic colloids: boundary conditions and penetration depth

Spherical colloids that catalyze the interconversion reaction A⇋B between solute molecules A and B whose concentration at infinity is maintained away from equilibrium effectively interact due to the non-uniform fields of solute concentrations. We show that this long range 1/r interaction is suppressed via a mechanism that is superficially reminiscent but qualitatively very different from electrostatic screening: catalytic activity drives the concentrations of solute molecules towards their equilibrium values and reduces the chemical imbalance that drives the interaction between the colloids. The imposed non-equilibrium boundary conditions give rise to a variety of geometry-dependent scenarios; while long-range interactions are suppressed (except for a finite penetration depth) in the bulk of the colloid solution in 3D, they can persist in quasi-2D geometry in which the colloids but not the solutes are confined to a surface, resulting in the formation of clusters or Wigner crystals, depending on the sign of the interaction between colloids.

Exact Phoretic Interaction of Two Chemically Active Particles

We study the nonequilibrium interaction of two isotropic chemically active particles taking into account the exact near-field chemical interactions as well as hydrodynamic interactions. We identify regions in the parameter space wherein the dynamical system describing the two particles can have a fixed point-a phenomenon that cannot be captured under the far-field approximation. We find that, due to near-field effects, the particles may reach a stable equilibrium at a nonzero gap size or make a complex that can dissociate in the presence of sufficiently strong noise. We explicitly show that the near-field effects originate from a self-generated neighbor-reflected chemical gradient, similar to interactions of a self-propelling phoretic particle and a flat substrate.

Universal aspects of collective behavior in chemotactic systems

We investigate the collective dynamics of particles (e.g., microorganisms) interacting via chemotactic gradients. Specifically, we focus on continuum models for chemotaxis that include a damping of the chemical production with increasing local particle density and/or systems where the chemotactic sensitivity is reduced with increasing local concentration of the chemical. Using a recently introduced perturbative method [Phys. Rev. E 98, 020603 (2018)10.1103/PhysRevE.98.020603], we show that the onset of particle clustering in these systems is described by the universal Cahn-Hilliard (CH) model. On the one hand, this establishes particle-conserving models for chemotaxis as a further example for the universal class of nonequilibrium demixing phenomena we call active phase separation. On the other hand, the reduction to the CH model allows an analytical determination of suitable parameter ranges wherein, e.g., the transition to spatial density modulations is continuous and/or undesired blow-up solutions can be avoided. A comparison between the numerical solutions of the chemotaxis model and the derived CH model also provides the parameter range where the basic chemotaxis model behaves like other systems showing active phase separation, including the coarsening behavior in two spatial dimensions. Our approach highlights how basic principles of pattern formation theory allow the identification of common basic properties in different chemotaxis models.

Dynamics near planar walls for various model self-phoretic particles

For chemically active particles suspended in a liquid solution and moving by self-phoresis, the dynamics near chemically inert, planar walls is studied theoretically by employing various choices for the activity function, i.e., the spatial distribution of the sites where various chemical reactions take place. We focus on the case of solutions composed of electrically neutral species. This analysis extends previous studies of the case that the chemical activity can be modeled effectively as the release of a "product" molecular species from parts of the surface of the particle by accounting for annihilation of the product molecules by chemical reactions, either on the rest of the surface of the particle or in the volume of the surrounding solution. We show that, for the models considered here, the emergence of "sliding" and "hovering" wall-bound states is a generic, robust feature. However, the details of these states, such as the range of parameters within which they occur, depend on the specific model for the activity function. Additionally, in certain cases there is a reversal of the direction of the motion compared to the one observed if the particle is far away from the wall. We have also studied the changes of the dynamics induced by a direct interaction between the particle and the wall by including a short-ranged repulsive component to the interaction in addition to the steric one (a procedure often employed in numerical simulations of active colloids). Upon increasing the strength of this additional component, while keeping its range fixed, significant qualitative changes occur in the phase portraits of the dynamics near the wall: for sufficiently strong short-ranged repulsion, the sliding steady states of the dynamics are transformed into hovering states. Furthermore, our studies provide evidence for an additional "oscillatory" wall-bound steady state of motion for chemically active particles due to a strong, short-ranged, and direct repulsion. This kind of particle translates along the wall at a distance from it which oscillates between a minimum and a maximum.

Active Phase Separation in Mixtures of Chemically Interacting Particles

We theoretically study mixtures of chemically interacting particles, which produce or consume a chemical to which they are attracted or repelled, in the most general case of many coexisting species. We find a new class of active phase separation phenomena in which the nonequilibrium chemical interactions between particles, which break action-reaction symmetry, can lead to separation into phases with distinct density and stoichiometry. Because of the generic nature of our minimal model, our results shed light on the underlying fundamental principles behind nonequilibrium self-organization of cells and bacteria, catalytic enzymes, or phoretic colloids.

Light-controlled assembly of active colloidal molecules

Thanks to a constant energy input, active matter can self-assemble into phases with complex architectures and functionalities such as living clusters that dynamically form, reshape, and break-up, which are forbidden in equilibrium materials by the entropy maximization (or free energy minimization) principle. The challenge to control this active self-assembly has evoked widespread efforts typically hinging on engineering of the properties of individual motile constituents. Here, we provide a different route, where activity occurs as an emergent phenomenon only when individual building blocks bind together in a way that we control by laser light. Using experiments and simulations of two species of immotile microspheres, we exemplify this route by creating active molecules featuring a complex array of behaviors, becoming migrators, spinners, and rotators. The possibility to control the dynamics of active self-assembly via light-controllable nonreciprocal interactions will inspire new approaches to understand living matter and to design active materials.

Magnetite nanoparticles program the assembly, response, and reconfiguration of structured emulsions

Endoskeletal droplets-non-spherical emulsion droplets that respond to external stimuli with shape change-are modified with ferromagnetic iron oxide nanoparticles to make them susceptible to magnetic fields. The resulting droplets can be manipulated using static or oscillating magnetic fields, each producing a different response. Static fields control the orientation and position of the droplets, important in driving assembly into organized structures. Oscillating fields are shown to cause magnetic hyperthermia in ferrofluid nanoparticles, leading to droplet heating and forcing droplet reconfiguration. We demonstrate the use of static and dynamic fields to affect the organization and stability of endoskeletal droplets and their assemblies, producing highly-tunable programmable colloids that adapt to changing environmental conditions.

Shape-directed rotation of homogeneous micromotors via catalytic self-electrophoresis

The pursuit of chemically-powered colloidal machines requires individual components that perform different motions within a common environment. Such motions can be tailored by controlling the shape and/or composition of catalytic microparticles; however, the ability to design particle motions remains limited by incomplete understanding of the relevant propulsion mechanism(s). Here, we demonstrate that platinum microparticles move spontaneously in solutions of hydrogen peroxide and that their motions can be rationally designed by controlling particle shape. Nanofabricated particles with n-fold rotational symmetry rotate steadily with speed and direction specified by the type and extent of shape asymmetry. The observed relationships between particle shape and motion provide evidence for a self-electrophoretic propulsion mechanism, whereby anodic oxidation and cathodic reduction occur at different rates at different locations on the particle surface. We develop a mathematical model that explains how particle shape impacts the relevant electrocatalytic reactions and the resulting electrokinetic flows that drive particle motion.

Self-propelled motors operating at the micro- or nanoscale can be powered by catalytic reactions and show appealing potential in robotic applications. Brooks et al. describe how the motions of platinum spinners in hydrogen peroxide solutions can be rationally designed by controlling particle shape.

Dissipative phase transitions in systems with nonreciprocal effective interactions

The reciprocity of effective interparticle forces can be violated in various open and nonequilibrium systems, in particular, in colloidal suspensions and complex (dusty) plasmas. Here, we obtain a criterion under which a nonreciprocal system can be strictly reduced to a pseudo-Hamiltonian system with a detailed dynamic equilibrium. In particular, the criterion is satisfied for catalytically active colloids interacting via nonreciprocal diffusiophoretic forces. However, in the general case, when this criterion is not satisfied, the steady state is determined by the interplay between dissipation and the energy source due to the nonreciprocity of interactions. The results indicate the realization of bistability and dissipative spinodal decomposition in a broad class of systems with nonreciprocal effective interactions.

Modular approach to microswimming

The field of active matter in general and microswimming in particular has experienced a rapid and ongoing expansion over the last decade. A particular interesting aspect is provided by artificial autonomous microswimmers constructed from individual active and inactive functional components into self-propelling complexes. Such modular microswimmers may exhibit directed motion not seen for each individual component. In this review, we focus on the establishment and recent developments in the modular approach to microswimming. We introduce the bound and dynamic prototypes, show mechanisms and types of modular swimming and discuss approaches to control the direction and speed of modular microswimmers. We conclude by highlighting some challenges faced by researchers as well as promising directions for future research in the realm of modular swimming.

Catalytic Janus Colloids: Controlling Trajectories of Chemical Microswimmers

Catalytic Janus colloids produce rapid motion in fluids by decomposing dissolved fuel. There is great potential to exploit these "autonomous chemical swimmers" in applications currently performed by diffusion limited passive colloids. Key application areas for colloids include transporting active ingredients for drug delivery, gathering analytes for medical diagnostics, and self-assembling into regular structures used for photonic materials and lithographic templating. For drug delivery and medical diagnostics, controlling colloidal motion is key in order to target therapies, and transport analytes through lab-on-a-chip devices. Here, the autonomous motion of catalytic Janus colloids can remove the current requirements to induce and control colloid motion using external fields, thereby reducing the technological complexity required for medical therapies and diagnostics. For materials applications exploiting colloidal self-assembly, the additional interactions introduced by catalytic activity and rapid motion are predicted to allow access to new reconfigurable and responsive structures. In order to realize these goals, it is vital to develop methods to control both individual colloidal paths and collective behavior in motile catalytic colloidal systems. However, catalytic Janus colloids' trajectories are randomized by Brownian effects, and so require new strategies in order to be harnessed for transport. This is achievable using a variety of different approaches. For example, self-assembly and control of catalyst geometry can introduce controlled amounts of rotary motion, or "spin" into chemical swimmer trajectories. Furthermore, rotary motion combined with gravity, produces well-defined orientated helical trajectories. In addition, when catalytic colloids interact with topographical features, such as edges and trenches, they are steered. This gives rise to a new approach for autonomous colloidal microfluidic transport that could be deployed in future lab-on-a-chip devices. Chemical gradients can also influence the motion of catalytic Janus colloids, for example, to cause collective accumulations at specific locations. However, at present, the predicted theoretical degree of control over this phenomenon has not been fully verified in experimental systems. Collective behavior control for chemical swimmers is also possible by exploiting the potential for the complex interactions in these systems to allow access to self-assembled, dynamic and reconfigurable ordered structures. Again, current experiments have not yet accessed the breadth of possible behavior. Consequently, continued efforts are required to understand and control these interaction mechanisms in real world systems. Ultimately, this will help realize the use of catalytic Janus colloids for tasks that require well-controlled motion and structural organization, enabling functions such as analyte capture and concentration, or targeted drug delivery.

Clustering-induced self-propulsion of isotropic autophoretic particles

Self-diffusiophoretic particles exploit local concentration gradients of a solute species in order to self-propel at the micron scale. While an isolated chemically- and geometrically-isotropic particle cannot swim, we show that it can achieve self-propulsion through interactions with other individually-non-motile particles by forming geometrically-anisotropic clusters via phoretic and hydrodynamic interactions. This result identifies a new route to symmetry-breaking for the concentration field and to self-propulsion, that is not based on an anisotropic design, but on the collective dynamics of identical and homogeneous active particles. Using full numerical simulations as well as theoretical modelling of the clustering process, the statistics of the propulsion properties are obtained for arbitrary initial arrangement of the particles. The robustness of these results to thermal noise, and more generally the effect of Brownian motion of the particles, is also discussed.

Vortex dynamics and Lagrangian statistics in a model for active turbulence

Cellular suspensions such as dense bacterial flows exhibit a turbulence-like phase under certain conditions. We study this phenomenon of "active turbulence" statistically by using numerical tools. Following Wensink et al. (Proc. Natl. Acad. Sci. U.S.A. 109, 14308 (2012)), we model active turbulence by means of a generalized Navier-Stokes equation. Two-point velocity statistics of active turbulence, both in the Eulerian and the Lagrangian frame, is explored. We characterize the scale-dependent features of two-point statistics in this system. Furthermore, we extend this statistical study with measurements of vortex dynamics in this system. Our observations suggest that the large-scale statistics of active turbulence is close to Gaussian with sub-Gaussian tails.

Generic Long-Range Interactions Between Passive Bodies in an Active Fluid

A single nonspherical body placed in an active fluid generates currents via breaking of time-reversal symmetry. We show that, when two or more passive bodies are placed in an active fluid, these currents lead to long-range interactions. Using a multipole expansion, we characterize their leading-order behaviors in terms of single-body properties and show that they decay as a power law with the distance between the bodies, are anisotropic, and do not obey an action-reaction principle. The interactions lead to rich dynamics of the bodies, illustrated by the spontaneous synchronized rotation of pinned nonchiral bodies and the formation of traveling bound pairs. The occurrence of these phenomena depends on tunable properties of the bodies, thus opening new possibilities for self-assembly mediated by active fluids.

Rational design and dynamics of self-propelled colloidal bead chains: from rotators to flagella

The quest for designing new self-propelled colloids is fuelled by the demand for simple experimental models to study the collective behaviour of their more complex natural counterparts. Most synthetic self-propelled particles move by converting the input energy into translational motion. In this work we address the question if simple self-propelled spheres can assemble into more complex structures that exhibit rotational motion, possibly coupled with translational motion as in flagella. We exploit a combination of induced dipolar interactions and a bonding step to create permanent linear bead chains, composed of self-propelled Janus spheres, with a well-controlled internal structure. Next, we study how flexibility between individual swimmers in a chain can affect its swimming behaviour. Permanent rigid chains showed only active rotational or spinning motion, whereas longer semi-flexible chains showed both translational and rotational motion resembling flagella like-motion, in the presence of the fuel. Moreover, we are able to reproduce our experimental results using numerical calculations with a minimal model, which includes full hydrodynamic interactions with the fluid. Our method is general and opens a new way to design novel self-propelled colloids with complex swimming behaviours, using different complex starting building blocks in combination with the flexibility between them.

Fore-aft asymmetric flocking

We show that fore-aft asymmetry, a generic feature of living organisms and some active matter systems, can have a strong influence on the collective properties of even the simplest flocking models. Specifically, an arbitrarily weak asymmetry favoring front neighbors changes qualitatively the phase diagram of the Vicsek model. A region where many sharp traveling band solutions coexist is present at low noise strength, below the Toner-Tu liquid, at odds with the phase-separation scenario well describing the usual isotropic model. Inside this region, a "banded-liquid" phase with algebraic density distribution coexists with band solutions. Linear stability analysis at the hydrodynamic level suggests that these results are generic and not specific to the Vicsek model.

Self-Assembly of Colloidal Molecules due to Self-Generated Flow

The emergence of structure through aggregation is a fascinating topic and of both fundamental and practical interest. Here we demonstrate that self-generated solvent flow can be used to generate long-range attractions on the colloidal scale, with subpiconewton forces extending into the millimeter range. We observe a rich dynamic behavior with the formation and fusion of small clusters resembling molecules. The dynamics of this assembly is governed by an effective conservative energy that for large separations r decays as 1/r. Breaking the flow symmetry, these clusters can be made active.

Hybrid colloidal microswimmers through sequential capillary assembly

Active colloids, also known as artificial microswimmers, are self-propelled micro- and nanoparticles that convert uniform sources of fuel (e.g. chemical) or uniform external driving fields (e.g. magnetic or electric) into directed motion by virtue of asymmetry in their shape or composition. These materials are currently attracting enormous scientific attention as models for out-of-equilibrium systems and with the promise to be used as micro- and nanoscale devices. However, current fabrication of active colloids is limited in the choice of available materials, geometries, and modes of motion. Here, we use sequential capillarity-assisted particle assembly (sCAPA) to link microspheres of different materials into hybrid clusters of prescribed shapes ("colloidal molecules") that can actively translate, circulate and rotate powered by asymmetric electro-hydrodynamic flows. We characterize the active motion of the clusters and highlight the range of parameters (composition and shape) that can be used to tune their trajectories. Further engineering provides active colloids that switch motion under external triggers or perform simple pick-up and transport tasks. By linking their design, realization and characterization, our findings enable and inspire both physicists and engineers to create customized active colloids to explore novel fundamental phenomena in active matter and to investigate materials and propulsion schemes that are compatible with future applications.

Chemically active colloids near osmotic-responsive walls with surface-chemistry gradients

Chemically active colloids move by creating gradients in the composition of the surrounding solution and by exploiting the differences in their interactions with the various molecular species in solution. If such particles move near boundaries, e.g. the walls of the container confining the suspension, gradients in the composition of the solution are also created along the wall. This give rise to chemi-osmosis (via the interactions of the wall with the molecular species forming the solution), which drives flows coupling back to the colloid and thus influences its motility. Employing an approximate 'point-particle' analysis, we show analytically that-owing to this kind of induced active response (chemi-osmosis) of the wall-such chemically active colloids can align with, and follow, gradients in the surface chemistry of the wall. In this sense, these artificial 'swimmers' exhibit a primitive form of thigmotaxis with the meaning of sensing the proximity of a (not necessarily discontinuous) physical change in the environment. We show that the alignment with the surface-chemistry gradient is generic for chemically active colloids as long as they exhibit motility in an unbounded fluid, i.e. this phenomenon does not depend on the exact details of the propulsion mechanism. The results are discussed in the context of simple models of chemical activity, corresponding to Janus particles with 'source' chemical reactions on one half of the surface and either 'inert' or 'sink' reactions over the other half.

‘Fuelled’ motion: phoretic motility and collective behaviour of active colloids

Phoretic motility is a propulsion mechanism used to design active particles which display complex collective behaviours, characterised experimentally and theoretically.

Fission and fusion scenarios for magnetic microswimmer clusters

Fission and fusion processes of particle clusters occur in many areas of physics and chemistry from subnuclear to astronomic length scales. Here we study fission and fusion of magnetic microswimmer clusters as governed by their hydrodynamic and dipolar interactions. Rich scenarios are found that depend crucially on whether the swimmer is a pusher or a puller. In particular a linear magnetic chain of pullers is stable while a pusher chain shows a cascade of fission (or disassembly) processes as the self-propulsion velocity is increased. Contrarily, magnetic ring clusters show fission for any type of swimmer. Moreover, we find a plethora of possible fusion (or assembly) scenarios if a single swimmer collides with a ringlike cluster and two rings spontaneously collide. Our predictions are obtained by computer simulations and verifiable in experiments on active colloidal Janus particles and magnetotactic bacteria.

Structural correlations in diffusiophoretic colloidal mixtures with nonreciprocal interactions

Nonreciprocal effective interaction forces can occur between mesoscopic particles in colloidal suspensions that are driven out of equilibrium. These forces violate Newton's third law actio  =  reactio on coarse-grained length and time scales. Here we explore the statistical mechanics of Brownian particles with nonreciprocal effective interactions. Our model system is a binary fluid mixture of spherically symmetric, diffusiophoretic mesoscopic particles, and we focus on the time-averaged particle pair- and triplet-correlation functions. Based on the many-body Smoluchowski equation we develop a microscopic statistical theory for the particle correlations and test it by computer simulations. For model systems in two and three spatial dimensions, we show that nonreciprocity induces distinct nonequilibrium pair correlations. Our predictions can be tested in experiments with chemotactic colloidal suspensions.

Active dipole clusters: From helical motion to fission

The structure of a finite particle cluster is typically determined by total energy minimization. Here we consider the case where a cluster of soft-sphere dipoles becomes active, i.e., when the individual particles exhibit an additional self-propulsion along their dipole moments. We numerically solve the overdamped equations of motion for soft-sphere dipoles in a solvent. Starting from an initial metastable dipolar cluster, the self-propulsion generates a complex cluster dynamics. The final cluster state has in general a structure widely different to the initial one, the details depend on the model parameters and on the protocol of how the self-propulsion is turned on. The center of mass of the cluster moves on a helical path, the details of which are governed by the initial cluster magnetization. An instantaneous switch to a high self-propulsion leads to fission of the cluster. However, fission does not occur if the self-propulsion is increased slowly to high strengths. Our predictions can be verified through experiments with self-phoretic colloidal Janus particles and for macroscopic self-propelled dipoles in a highly viscous solvent.