-The structural and dynamical properties of suspensions of self-propelled Brownian particles of spherical shape are investigated in three spatial dimensions. Our simulations reveal a phase separation into a dilute and a dense phase, above a certain density and strength of selfpropulsion. The packing fraction of the dense phase approaches random close packing at high activity, yet the system remains fluid. Although no alignment mechanism exists in this model, we find long-lived cooperative motion of the particles in the dense regime. This behavior is probably due to an interface-induced sorting process. Spatial displacement correlation functions are nearly scale-free for systems with densities close to or above the glass transition density of passive systems.

The pressure of suspensions of self-propelled objects is studied theoretically and by simulation of spherical active Brownian particles (ABPs). We show that for certain geometries, the mechanical pressure as force/area of confined systems can be equally expressed by bulk properties, which implies the existence of a nonequilibrium equation of state. Exploiting the virial theorem, we derive expressions for the pressure of ABPs confined by solid walls or exposed to periodic boundary conditions. In both cases, the pressure comprises three contributions: the ideal-gas pressure due to white-noise random forces, an activity-induced pressure ("swim pressure"), which can be expressed in terms of a product of the bare and a mean effective particle velocity, and the contribution by interparticle forces. We find that the pressure of spherical ABPs in confined systems explicitly depends on the presence of the confining walls and the particle-wall interactions, which has no correspondence in systems with periodic boundary conditions. Our simulations of three-dimensional ABPs in systems with periodic boundary conditions reveal a pressure-concentration dependence that becomes increasingly nonmonotonic with increasing activity. Above a critical activity and ABP concentration, a phase transition occurs, which is reflected in a rapid and steep change of the pressure. We present and discuss the pressure for various activities and analyse the contributions of the individual pressure components.

A variety of computational models have been developed to describe active matter at different length and time scales. The diversity of the methods and the challenges in modeling active matterranging from molecular motors and cytoskeletal filaments over artificial and biological swimmers on microscopic to groups of animals on macroscopic scales-mainly originate from their out-ofequilibrium character, multiscale nature, nonlinearity, and multibody interactions. In the present review, various modeling approaches and numerical techniques are addressed, compared, and differentiated to illuminate the innovations and current challenges in understanding active matter. The complexity increases from minimal microscopic models of dry active matter toward microscopic models of active matter in fluids. Complementary, coarse-grained descriptions and continuum models are elucidated. Microscopic details are often relevant and strongly affect collective behaviors, which implies that the selection of a proper level of modeling is a delicate choice, with simple models emphasizing universal properties and detailed models capturing specific features. Finally, current approaches to further advance the existing models and techniques to cope with real-world applications, such as complex media and biological environments, are discussed.2

We present quantitative experimental data on colloidal laning at the single-particle level. Our results demonstrate a continuous increase in the fraction of particles in a lane for the case where oppositely charged particles are driven by an electric field. This behavior is accurately captured by Brownian dynamics simulations. By studying the fluctuations parallel and perpendicular to the field we identify the mechanism that underlies the formation of lanes.Far from thermodynamic equilibrium, a wealth of fascinating selforganization processes can emerge along with unusual pattern formation and novel transport properties. 1,2 One of the simplest prototypes of non-equilibrium pattern formation is lane formation, exhibited by dusty plasmas, 3,4 granular matter, 5 pedestrian dynamics, 6,7 and army ants. 8 In this paper, we characterize the patterning and dynamical signatures of lane formation in a colloidal system experimentally and with computer simulations. Our results may find use in electronic ink, which also contains oppositely charged colloids that are driven by electric fields. 9,10 A fundamental and microscopic understanding of non-equilibrium phenomena requires resolving the underlying dynamical processes on the scale of the individual particles. For this, colloidal dispersions are excellent model systems since they can be brought out of equilibrium in a controlled way by external fields and the trajectories of the individual particles can be tracked in real space using confocal microscopy, which allows unparalleled comparison with computer simulation and particle-level theory. 11-13 Here, we first study the formation of lanes in driven colloidal mixtures as a function of the driving strength using both experiments and Brownian dynamics computer simulations. Lane formation in this 3D system is found to be a continuous process as a function of driving field. Starting from an initial mixed state, the dynamical mechanism behind the formation of lanes is identified: there is an enhanced lateral mobility of particles induced by collisions with particles driven in the opposite direction, which sharply decreases once lanes are formed. Therefore, particles in a lane can be regarded as being in a dynamically 'locked-in' state.In our experiments, we used a binary dispersion of sterically stabilized, nearly equal sized, but oppositely charged polymethylmethacrylate (PMMA) spheres inside a rectangular capillary. The particles were synthesized by dispersion polymerization, 14 and fluorescently labeled with either 7-nitrobenzo-2-oxa-1,3-diazole (NBD) or rhodamine isothiocyanate (RITC). The two species are color-coded as 'green' (s green ¼ 1.06 mm, polydispersity 6%, NBDlabeled) and 'red' (s red ¼ 0.91 mm, polydispersity 7%, RITC-labeled). The overall volume fraction of the suspension f green (0.090) + f red (0.090) was 0.18.To match the density and refractive index of the particles with the solvent, the particles were dispersed in a mixture of 27.2 w% cisdecahydronaphthalene and cyclohexylbromide containing 75 mM tetrabut...

Abstract. We show how to implement stick boundary conditions for a spherical colloid in a solvent that is coarse-grained by the method of stochastic rotation dynamics. This allows us to measure colloidal rotational velocity auto-correlation functions by direct computer simulation. We find quantitative agreement with Enskog theory for short times and with hydrodynamic mode-coupling theory for longer times. For aqueous colloidal suspensions, the Enskog contribution to the rotational friction is larger than the hydrodynamic one when the colloidal radius drops below 35nm.

A mesoscopic hydrodynamic model to simulate synthetic self-propelled Janus particles which is thermophoretically or diffusiophoretically driven is here developed. We first propose a model for a passive colloidal sphere which reproduces the correct rotational dynamics together with strong phoretic effect. This colloid solution model employs a multiparticle collision dynamics description of the solvent, and combines potential interactions with the solvent, with stick boundary conditions. Asymmetric and specific colloidal surface is introduced to produce the properties of self-phoretic Janus particles. A comparative study of Janus and microdimer phoretic swimmers is performed in terms of their swimming velocities and induced flow behavior. Self-phoretic microdimers display long range hydrodynamic interactions and can be characterized as pullers or pushers. In contrast, Janus particles are characterized by short range hydrodynamic interactions and behave as neutral swimmers. Our model nicely mimics those recent experimental realization of the self-phoretic Janus particles.

Bacteria such as Escherichia coli swim along circular trajectories adjacent to surfaces. Thereby, the orientation (clockwise, counterclockwise) and the curvature depend on the surface properties. We employ mesoscale hydrodynamic simulations of a mechano-elastic model of E. coli, with a spherocylindrical body propelled by a bundle of rotating helical flagella, to study quantitatively the curvature of the appearing circular trajectories. We demonstrate that the cell is sensitive to nanoscale changes in the surface slip length. The results are employed to propose a novel approach to directing bacterial motion on striped surfaces with different slip lengths, which implies a transformation of the circular motion into a snaking motion along the stripe boundaries. The feasibility of this approach is demonstrated by a simulation of active Brownian rods, which also reveals a dependence of directional motion on the stripe width.

The dynamical onset of lane formation is studied in experiments with binary complex plasmas under microgravity conditions. Small microparticles are driven and penetrate into a cloud of big particles, revealing a strong tendency towards lane formation. The observed time-resolved lane-formation process is in good agreement with computer simulations of a binary Yukawa model with Langevin dynamics. The laning is quantified in terms of the anisotropic scaling index, leading to a universal order parameter for driven systems.