Dynamical Crystallites of Active Chiral Particles

Emergent mesoscale correlations in active solids with noisy chiral dynamics

We present the linear response theory for an elastic solid composed of active Brownian particles with intrinsic individual chirality, deriving both a normal mode formulation and a continuum elastic formulation. Using this theory, we compute analytically the velocity correlations and energy spectra under different conditions, showing an excellent agreement with simulations. We generate the corresponding phase diagram, identifying chiral and achiral disordered regimes (for high chirality or noise levels), as well as chiral and achiral states with mesoscopic-range order (for low chirality and noise). The chiral ordered states display mesoscopic spatial correlations and oscillating time correlations, but no wave propagation. In the high chirality regime, we find a peak in the elastic energy spectrum that leads to a non-monotonic behavior with increasing noise strength that is consistent with the emergence of the 'hammering state' recently identified in chiral glasses. Finally, we show numerically that our theory, despite its linear response nature, can be applied beyond the idealized homogeneous solid assumed in our derivations. Indeed, by increasing the level of activity, we show that it remains a good approximation of the system dynamics until just below the melting transition. In addition, we show that there is still an excellent agreement between our analytical results and simulations when we extend our results to heterogeneous solids composed of mixtures of active particles with different intrinsic chirality and noise levels. The derived linear response theory is therefore robust and applicable to a broad range of real-world active systems. Our work provides a thorough analytical and numerical description of the emergent states in a densely packed system of chiral self-propelled Brownian disks, thus allowing a detailed understanding of the phases and dynamics identified in a minimal chiral active system.

Collective motion of binary chiral particle mixtures with environmental complex noise

The strategies for demixing and sorting mixed-chirality particles are crucial in the biochemical and pharmaceutical industries. However, whether chiral mixed particles can effectively separate in more complex spatial environments remains unresolved. In this paper, we explore the collective motion of binary chiral particle mixtures with environmental complex noise in the binary chiral Vicsek model (BCVM). We discover that the noisy region ratio, λ, significantly influences the separation behavior and spatial distribution of binary mixtures, unveiling system states not observed in uniform environments. Additionally, varying the chirality of particles reveals four distinct phases in our model. In the Vicsek bands phase (small chirality), an increase in λ can, under certain conditions, promote segregation rather than consistently hindering the demixing process. Conversely, for large chirality, localized dynamics and a homogeneous phase emerge, reducing the impact of λ on separation behavior. Notably, when chirality and activity are comparable, macrodrops and microflock phases appear, with a mixed-segregated state transition occurring at a critical λ_{c}. For λ<λ_{c}, chiral separation occurs with particles confined to the noise-free region. However, when λ>λ_{c}, particles gradually migrate to the noisy region, disrupting the separation. Further, we discuss the effects of multiple factors, including chirality, velocity, noise magnitude, particle number, and system size on λ_{c}. We also identify an optimal interaction radius at which λ_{c} reaches its peak value. Our findings may inspire strategies for achieving spontaneous demixing and spatial migration of mixed-chirality particles in complex environments.

Continuous symmetry breaking of low-dimensional systems driven by inhomogeneous oscillatory driving forces

The driving forces of chiral active particles and deformations of cells are often modeled by spatially inhomogeneous but temporally periodic driving forces. Such inhomogeneous oscillatory driving forces have only recently been proposed in the context of active matter, and their effects on the systems are not yet fully understood. In this work, we theoretically study the impact of spatially inhomogeneous oscillatory driving forces on continuous symmetry breaking. We first analyze the linear model for the soft modes in the ordered phase to derive the lower critical dimension of the model, and then analyze the spherical model to investigate more detailed phase behaviors. Interestingly, our analysis reveals that symmetry breaking occurs even in one and two dimensions, where the Hohenberg-Mermin-Wagner theorem prohibits continuous symmetry breaking in equilibrium. Furthermore, fluctuations of conserved quantities, such as density, are anomalously suppressed in the long-wavelength, i.e., show hyperuniformity.

Collective Hall current in chiral active fluids: Coupling of phase and mass transport through traveling bands

Active fluids composed of constituents that are constantly driven away from thermal equilibrium can support spontaneous currents and can be engineered to have unconventional transport properties. Here, we report the emergence of (meta)stable traveling bands in computer simulations of aligning circle swimmers. These bands are different from polar flocks and, through coupling phase with mass transport, induce a bulk particle current with a component perpendicular to the propagation direction, thus giving rise to a collective Hall (or Magnus) effect. Traveling bands require sufficiently small orbits and undergo a discontinuous transition into a synchronized state with transient polar clusters for large orbital radii. Within a minimal hydrodynamic theory, we show that the bands can be understood as nondispersive soliton solutions fully accounting for the numerically observed properties.

Stability of the chiral crystal phase and breakdown of the cholesteric phase in mixtures of active-passive chiral rods

In this study, we aim to explore the effect of chirality on the phase behavior of active helical particles driven by two-temperature scalar activity. We first calculate the equation of state of soft helical particles of various intrinsic chiralities using molecular dynamics (MD) simulation. In equilibrium, the emergence of various liquid crystal (LC) phases such as nematic (N), cholesteric , smectic (Sm) and crystal (K) crucially depends on the presence of walls that induce planar alignment. Next, we introduce activity through the two-temperature model: keep increasing the temperature of half of the helical particles (labeled as 'hot' particles) while maintaining the temperature of the other half at a lower value (labeled as 'cold' particles). Starting from a homogeneous isotropic (I) phase, we find the emergence of 2-TIPS: two temperature-induced phase separations between the hot and cold particles. We also observe that the cold particles undergo an ordering transition to various LC phases even in the absence of a wall. This observation reveals that the hot-cold interface in the active system plays the role of a wall in the equilibrium system by inducing an alignment direction for the cold particles. However, in the case of a cholesteric phase, we observe that activity destabilizes the phase by inducing smectic ordering in the cold zone while an isotropic structure in the hot zone. The smectic ordering in the cold zone eventually transforms to a chiral crystal phase with high enough activity.

Collective motion of chiral particles in complex noise environments

Collective motion of chiral particles in complex noise environments is investigated based on the Vicsek model. In the model, we added chirality, along with complex noise, affecting particles clustering motion. Particles can only avoid noise interference in a specific channel, and this consideration is more realistic due to the complexity of the environment. Via simulations, we find that the channel proportion, p, critically influences chiral particle synchronization. Specifically, we observe a disorder-order transition at critical [Formula: see text], only when [Formula: see text], the system can achieve global synchronization. Combined with our definition of spatial distribution parameter and observation of the model, the reason is that particles begin to escape from the noise region under the influence of complex noise. In addition, the value of [Formula: see text] increases linearly with velocity, while it decreases monotonically with the increase in chirality and interaction radius. Interestingly, an appropriate noise amplitude minimizes [Formula: see text]. Our findings may inspire novel strategies to manipulate self-propelled particles of distinct chirality to achieve desired spatial migration and global synchronization.

Active smectics on a sphere

The dynamics of active smectic liquid crystals confined on a spherical surface is explored through an active phase field crystal model. Starting from an initially randomly perturbed isotropic phase, several types of topological defects are spontaneously formed, and then annihilate during a coarsening process until a steady state is achieved. The coarsening process is highly complex involving several scaling laws of defect densities as a function of time where different dynamical exponents can be identified. In general the exponent for the final stage towards the steady state is significantly larger than that in the passive and in the planar case, i.e. the coarsening is getting accelerated both by activity and by the topological and geometrical properties of the sphere. A defect type characteristic for this active system is a rotating spiral of evolving smectic layering lines. On a sphere this defect type also determines the steady state. Our results can in principle be confirmed by dense systems of synthetic or biological active particles.

Spontaneous Chiral Symmetry Breaking and Lane Formation in Ferromagnetic Ferrofluids

Ferromagnetic ferrofluids are synthetic materials consisting of magnetic nanoplatelets dispersed in an isotropic fluid. Their main characteristics are the formation of stable magnetic domains and the presence of macroscopic magnetization even in the absence of a magnetic field. Here, the authors report on the experimental observation of spontaneous stripe formation in a ferromagnetic ferrofluid in the presence of an oscillating external magnetic field. The striped structure is identified as elongated magnetic domains, which exhibit reorientation upon reversal of the magnetic field. The stripes are oriented perpendicular to the magnetic field and are separated by alternating flow lanes. The velocity profile is measured using a space-time correlation technique that follows the motion of the thermally excited fluctuations in the sample. The highest velocities are found in the depleted regions between individual domains and reach values up to several µm s-1 . The fluid in adjacent lanes moves in the opposite directions despite the applied magnetic field being uniform. The formation of bidirectional flow lanes can be explained by alternating rotation of magnetic nanoparticles in neighboring stripes, which indicates spontaneous breaking of the chiral symmetry in the sample.

Entropy production and collective excitations of crystals out of equilibrium: The concept of entropons

We study the collective vibrational excitations of crystals under out-of-equilibrium steady conditions that give rise to entropy production. Their excitation spectrum comprises equilibriumlike phonons of thermal origin and additional collective excitations called entropons because each of them represents a mode of spectral entropy production. Entropons coexist with phonons and dominate them when the system is far from equilibrium while they are negligible in near-equilibrium regimes. The concept of entropons has been recently introduced and verified in a special case of crystals formed by self-propelled particles. Here we show that entropons exist in a broader class of active crystals that are intrinsically out of equilibrium and characterized by the lack of detailed balance. After a general derivation, several explicit examples are discussed, including crystals consisting of particles with alignment interactions and frictional contact forces.

Extreme Spontaneous Deformations of Active Crystals

We demonstrate that two-dimensional crystals made of active particles can experience extremely large spontaneous deformations without melting. Using particles mostly interacting via pairwise repulsive forces, we show that such active crystals maintain long-range bond order and algebraically decaying positional order, but with an exponent η not limited by the 1/3 bound given by the (equilibrium) KTHNY theory. We rationalize our findings using linear elastic theory and show the existence of two well-defined effective temperatures quantifying respectively large-scale deformations and bond-order fluctuations. The root of these phenomena lies in the sole time-persistence of the intrinsic axes of particles, and they should thus be observed in many different situations.

Self-Solidifying Active Droplets Showing Memory-Induced Chirality

Most synthetic microswimmers do not reach the autonomy of their biological counterparts in terms of energy supply and diversity of motions. Here, this work reports the first all-aqueous droplet swimmer powered by self-generated polyelectrolyte gradients, which shows memory-induced chirality while self-solidifying. An aqueous solution of surface tension-lowering polyelectrolytes self-solidifies on the surface of acidic water, during which polyelectrolytes are gradually emitted into the surrounding water and induce linear self-propulsion via spontaneous symmetry breaking. The low diffusion coefficient of the polyelectrolytes leads to long-lived chemical trails which cause memory effects that drive a transition from linear to chiral motion without requiring any imposed symmetry breaking. The droplet swimmer is capable of highly efficient removal (up to 85%) of uranium from aqueous solutions within 90 min, benefiting from self-propulsion and flow-induced mixing. These results provide a route to fueling self-propelled agents which can autonomously perform chiral motion and collect toxins.

Chiral active matter in external potentials

We investigate the interplay between chirality and confinement induced by the presence of an external potential. For potentials having radial symmetry, the circular character of the trajectories induced by the chiral motion reduces the spatial fluctuations of the particle, thus providing an extra effective confining mechanism, that can be interpreted as a lowering of the effective temperature. In the case of non-radial potentials, for instance, with an elliptic shape, chirality displays a richer scenario. Indeed, the chirality can break the parity symmetry of the potential that is always fulfilled in the non-chiral system. The probability distribution displays a strong non-Maxwell-Boltzmann shape that emerges in cross-correlations between the two Cartesian components of the position, that vanishes in the absence of chirality or when radial symmetry of the potential is restored. These results are obtained by considering two popular models in active matter, i.e. chiral Active Brownian particles and chiral active Ornstein-Uhlenbeck particles.

Entropons as collective excitations in active solids

The vibrational dynamics of solids is described by phonons constituting basic collective excitations in equilibrium crystals. Here, we consider a non-equilibrium active solid, formed by self-propelled particles, which bring the system into a non-equilibrium steady-state. We identify novel vibrational collective excitations of non-equilibrium (active) origin, which coexist with phonons and dominate over them when the system is far from equilibrium. These vibrational excitations are interpreted in the framework of non-equilibrium physics, in particular, stochastic thermodynamics. We call them "entropons" because they are the modes of spectral entropy production (at a given frequency and wave vector). The existence of entropons could be verified in future experiments on dense self-propelled colloidal Janus particles and granular active matter, as well as in living systems, such as dense cell monolayers.

Unidirectional rotation of micromotors on water powered by pH-controlled disassembly of chiral molecular crystals

Biological and synthetic molecular motors, fueled by various physical and chemical means, can perform asymmetric linear and rotary motions that are inherently related to their asymmetric shapes. Here, we describe silver-organic micro-complexes of random shapes that exhibit macroscopic unidirectional rotation on water surface through the asymmetric release of cinchonine or cinchonidine chiral molecules from their crystallites asymmetrically adsorbed on the complex surfaces. Computational modeling indicates that the motor rotation is driven by a pH-controlled asymmetric jet-like Coulombic ejection of chiral molecules upon their protonation in water. The motor is capable of towing very large cargo, and its rotation can be accelerated by adding reducing agents to the water.

Dynamics and interactions of Quincke roller clusters: From orbits and flips to excited states

Active matter systems may be characterized by the conversion of energy into active motion, e.g., the self-propulsion of microorganisms. Artificial active colloids form models that exhibit essential properties of more complex biological systems but are amenable to laboratory experiments. While most experimental models consist of spheres, active particles of different shapes are less understood. Furthermore, interactions between these anisotropic active colloids are even less explored. Here, we investigate the motion of active colloidal clusters and the interactions between them. We focus on self-assembled dumbbells and trimers powered by an external dc electric field. For dumbbells, we observe an activity-dependent behavior of spinning, circular, and orbital motions. Moreover, collisions between dumbbells lead to the hierarchical self-assembly of tetramers and hexamers, both of which form rotational excited states. On the other hand, trimers exhibit flipping motion that leads to trajectories reminiscent of a honeycomb lattice.

Glassy Dynamics in Chiral Fluids

Chiral active matter is enjoying a rapid increase of interest, spurred by the rich variety of asymmetries that can be attained in, e.g., the shape or self-propulsion mechanism of active particles. Though this has already led to the observance of so-called chiral crystals, active chiral glasses remain largely unexplored. A possible reason for this could be the naive expectation that interactions dominate the glassy dynamics and the details of the active motion become increasingly less relevant. Here, we show that quite the opposite is true by studying the glassy dynamics of interacting chiral active Brownian particles. We demonstrate that when our chiral fluid is pushed to glassy conditions, it exhibits highly nontrivial dynamics, especially compared to a standard linear active fluid such as common active Brownian particles. Despite the added complexity, we are still able to present a full rationalization for all identified dynamical regimes. Most notably, we introduce a new "hammering" mechanism, unique to rapidly spinning particles in high-density conditions, that can fluidize a chiral active solid.

Perspective: New directions in dynamical density functional theory

Classical dynamical density functional theory (DDFT) has become one of the central modeling approaches in nonequilibrium soft matter physics. Recent years have seen the emergence of novel and interesting fields of application for DDFT. In particular, there has been a remarkable growth in the amount of work related to chemistry. Moreover, DDFT has stimulated research on other theories such as phase field crystal models and power functional theory. In this perspective, we summarize the latest developments in the field of DDFT and discuss a variety of possible directions for future research.

Circular motion subject to external alignment under active driving: Nonlinear dynamics and the circle map

Hardly any real self-propelling or actively driven object is perfect. Thus, undisturbed motion will generally not follow straight lines but rather bent or circular trajectories. We here address self-propelled or actively driven objects that move in discrete steps and additionally tend to migrate towards a certain direction by discrete angular adjustment. Overreaction in the angular alignment is possible. This competition implies pronounced nonlinear dynamics including period doubling and chaotic behavior in a broad parameter regime. Such behavior directly affects the appearance of the trajectories. Furthermore, we address collective motion and effects of spatial self-concentration.

Percolation transition in phase-separating active fluid

The motility-induced phase separation exhibited by active particles with repulsive interactions is well known. We show that the interaction softness of active particles destabilizes the highly ordered dense phase, leading to the formation of a porous cluster which spans the system. This soft limit can also be achieved if the particle motility is increased beyond a critical value, at which the system clearly exhibits all the characteristics of a standard percolation transition. We also show that in the athermal limit, active particles exhibit similar transitions even at low motility. With these additional new phases, the phase diagram of repulsive active particles is revealed to be richer than what was previously conceived.

Biased-angle effect on diffusion dynamics and phase separation in anisotropic active particle system

A deep understanding for collective behavior in an active matter system with complex interactions has far-reaching impact in biology. In the present work, we adopt Langevin dynamics simulations to investigate diffusion dynamics and phase separation in an anisotropic active particle system with a tunable biased angle α defined as the deviation between the active force direction and anisotropic orientation. Our results demonstrate that the biased angle can induce super-rotational diffusion dynamics characterized by a power-law relationship between the mean square angle displacement (MSAD) and the time interval Δ t in the form of MSAD ∼ Δ t β with β > 1 and also result in non-trivial phase separation kinetics. As activity is dominant, nucleation time shows a non-monotonic dependence on the biased angle. Moreover, there arises a distinct transition of phase separation, from spinodal decomposition without apparent nucleation time to binodal decomposition with prominent nucleation delay. A significant inhibition effect occurs at right and obtuse angles, where the remarkable super-rotational diffusion prevents particle aggregation, leading to a slow nucleation process. As active force is competitive to anisotropic interactions, the system is almost homogeneous, while, intriguingly, we observe a re-entrant phase separation as a small acute angle is introduced. The prominent super-rotational diffusion under small angles provides an optimum condition for particle adsorption and cluster growth and, thus, accounts for the re-entrance of phase separation. A consistent scenario for the physical mechanism of our observations is achieved by properly considering the modulation of the biased angle on the interplay between activity and anisotropic interactions.

Deformable active nematic particles and emerging edge currents in circular confinements

We consider a microscopic field theoretical approach for interacting active nematic particles. With only steric interactions the self-propulsion strength in such systems can lead to different collective behaviour, e.g. synchronized self-spinning and collective translation. The different behaviour results from the delicate interplay between internal nematic structure, particle shape deformation and particle-particle interaction. For intermediate active strength an asymmetric particle shape emerges and leads to chirality and self-spinning crystals. For larger active strength the shape is symmetric and translational collective motion emerges. Within circular confinements, depending on the packing fraction, the self-spinning regime either stabilizes positional and orientational order or can lead to edge currents and global rotation which destroys the synchronized self-spinning crystalline structure.

Morphologies and dynamics of the interfaces between active and passive phases

Active matters exhibit interesting collective behaviors and novel phases, which provide an important platform for the study of nonequilibrium physics. Mixtures of active and passive particles have been intensively studied in motility-induced phase separation, but the morphology of the active-passive interface has been poorly explored. In this work, we investigate the interface morphology in two-dimensional mixtures of active and passive particles using Brownian dynamics simulations. By systematically changing the Péclet number (Pe) and area fraction (ρ), we obtain the phase diagram of the active-passive interface, including rough sharp, rough invasive and flat interdiffusive interfaces. For a sharp interface, dynamic scaling analysis in the propagation stage shows that the roughness exponent α, the growth exponent β, the time exponent κ, and the dynamic exponent z satisfy z = α/(β - κ). Such anomalous scaling indicates that the roughening behavior does not belong to the conventional universality classes with Family-Vicsek scaling for the growth of passive interfaces. On the other hand, the interface in the middle-wavelength regime during the morphology relaxation stage can be described by capillary wave theory. The mean interface position propagates with time as t^1/2, which is robust at different ρ and Pe values in the propagation stage and exhibits superdiffusion in the morphology relaxation stage. These similarities and differences between the active-inactive interfaces and passive interfaces cast light on the interfacial growth of active matter.

Emergent vortices and phase separation in systems of chiral active particles with dipolar interactions

Using Brownian dynamics (BD) simulations we investigate the self-organization of a monolayer of chiral active particles with dipolar interactions. Each particle is driven by both, translational and rotational self-propulsion, and carries a permanent point dipole moment at its center. The direction of the translational propulsion for each particle is chosen to be parallel to its dipole moment. Simulations are performed at high dipolar coupling strength and a density below that related to motility-induced phase separation in simple active Brownian particles. Despite this restriction, we observe a wealth of phenomena including formation of two types of vortices, phase separation, and flocking transitions. To understand the appearance and disappearance of vortices in the many-particle system, we further investigate the dynamics of simple ring structures under the impact of self-propulsion.

Two-dimensional localized states in an active phase-field-crystal model

The active phase-field-crystal (active PFC) model provides a simple microscopic mean field description of crystallization in active systems. It combines the PFC model (or conserved Swift-Hohenberg equation) of colloidal crystallization and aspects of the Toner-Tu theory for self-propelled particles. We employ the active PFC model to study the occurrence of localized and periodic active crystals in two spatial dimensions. Due to the activity, crystalline states can undergo a drift instability and start to travel while keeping their spatial structure. Based on linear stability analyses, time simulations, and numerical continuation of the fully nonlinear states, we present a detailed analysis of the bifurcation structure of resting and traveling states. We explore, for instance, how the slanted homoclinic snaking of steady localized states found for the passive PFC model is modified by activity. Morphological phase diagrams showing the regions of existence of various solution types are presented merging the results from all the analysis tools employed. We also study how activity influences the crystal structure with transitions from hexagons to rhombic and stripe patterns. This in-depth analysis of a simple PFC model for active crystals and swarm formation provides a clear general understanding of the observed multistability and associated hysteresis effects, and identifies thresholds for qualitative changes in behavior.

Surfactants and rotelles in active chiral fluids

Surfactant molecules migrate to interfaces, reduce interfacial tension, and form micelles. All of these behaviors occur at or near equilibrium. Here, we describe active analogs of surfactants that operate far from equilibrium in active chiral fluids. Unlike molecular surfactants, the amphiphilic character of surfactants in active chiral fluids is a consequence of their activity. Our fluid of choice is a mixture of spinners that demixes into left-handed and right-handed chiral fluid domains. We realize spinners in experiment with three-dimensionally printed vibrots. Vibrot surfactants are chains of vibrots containing both types of handedness. Experiments demonstrate the affinity of double-stranded chains to interfaces, where they glide along and act as mixing agents. Simulations access larger systems in which single-stranded chains form spinning vesicles, termed rotelles. Rotelles are the chiral analogs of micelles. Rotelle formation is a ratchet mechanism catalyzed by the vorticity of the chiral fluid and only exist far from equilibrium.