Non-equilibrium surface tension of the vapour-liquid interface of active Lennard-Jones particles

Phase Coexistence and Edge Currents in the Chiral Lennard-Jones Fluid

We study a model chiral fluid in two dimensions composed of Brownian disks interacting via a Lennard-Jones potential and a nonconservative transverse force, mimicking colloids spinning at a given rate. The system exhibits a phase separation between a chiral liquid and a dilute gas phase that can be characterized using a thermodynamic framework. We compute the equations of state and show that the surface tension controls interface corrections to the coexisting pressure predicted from the equal-area construction. Transverse forces increase surface tension and generate edge currents at the liquid-gas interface. The analysis of these currents shows that the rotational viscosity introduced in chiral hydrodynamics is consistent with microscopic bulk mechanical measurements. Chirality can also break the solid phase, giving rise to a dense fluid made of rotating hexatic patches. Our Letter paves the way for the development of the statistical mechanics of chiral particles assemblies.

Nonequilibrium currents in stochastic field theories: A geometric insight

We introduce a formalism to study nonequilibrium steady-state probability currents in stochastic field theories. We show that generalizing the exterior derivative to functional spaces allows identification of the subspaces in which the system undergoes local rotations. In turn, this allows prediction of the counterparts in the real, physical space of these abstract probability currents. The results are presented for the case of the Active Model B undergoing motility-induced phase separation, which is known to be out of equilibrium but whose steady-state currents have not yet been observed, as well as for the Kardar-Parisi-Zhang equation. We locate and measure these currents and show that they manifest in real space as propagating modes localized in regions with nonvanishing gradients of the fields.

Flocking without Alignment Interactions in Attractive Active Brownian Particles

Within a simple model of attractive active Brownian particles, we predict flocking behavior and challenge the widespread idea that alignment interactions are necessary to observe this collective phenomenon. Here, we show that even nonaligning attractive interactions can lead to a flocking state. Monitoring the velocity polarization as the order parameter, we reveal the onset of a first-order transition from a disordered phase, characterized by several small clusters, to a flocking phase, where a single flocking cluster is emerging. The scenario is confirmed by studying the spatial connected correlation function of particle velocities, which reveals scale-free behavior in flocking states and exponential-like decay for nonflocking configurations. Our predictions can be tested in microscopic and macroscopic experiments showing flocking, such as animals, migrating cells, and active colloids.

Morphologies and dynamics of free surfaces of crystals composed of active particles

Active matter exhibits various collective motions and nonequilibrium phases, such as crystals; however, their surface properties have been poorly explored. Here, we use Brownian dynamics simulations to investigate the surface morphology and dynamics of two-dimensional active crystals during and after growth. For crystal growth on a substrate, the position and roughness of the crystal surface reach steady states at different times. In the steady state, the surface exhibits superdiffusive behaviour at the short time, and the roughness is insensitive to the roughening process and particle activity. We observe two-stage and three-stage surface roughening at different Péclet numbers. The result of dynamic scaling analysis shows that the surface is similar to anomalous roughening, which is distinct from the normal roughening typically found in conventional passive systems. Capillary wave theory for a thermal equilibrium system can describe the active surface fluctuations only in the long-wavelength regime, indicating that active particles mainly drive the surface out of equilibrium locally. These similarities and differences between the active and passive crystal surfaces are essential for understanding active crystals and interfaces.

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.

Why Noether's theorem applies to statistical mechanics

Noether's theorem is familiar to most physicists due its fundamental role in linking the existence of conservation laws to the underlying symmetries of a physical system. Typically the systems are described in the particle-based context of classical mechanics or on the basis of field theory. We have recently shown (2021Commun. Phys.4176) that Noether's reasoning also applies to thermal systems, where fluctuations are paramount and one aims for a statistical mechanical description. Here we give a pedagogical introduction based on the canonical ensemble and apply it explicitly to ideal sedimentation. The relevant mathematical objects, such as the free energy, are viewed as functionals. This vantage point allows for systematic functional differentiation and the resulting identities express properties of both macroscopic average forces and molecularly resolved correlations in many-body systems, both in and out-of-equilibrium, and for active Brownian particles. To provide further background, we briefly describe the variational principles of classical density functional theory, of power functional theory, and of classical mechanics.

Competing effects of rotational diffusivity and activity on finite-sized clusters

Colloidal particles interacting via short-range attraction and long-range repulsion are known to stabilize finite-sized clusters under equilibrium conditions. In this work, the effect of self-propulsion speed (activity) and rotational diffusivity (D_r) on the phase behavior of such particles is investigated using Brownian dynamics simulations. The system exhibits rich phase behavior consisting of clusters of different kinds. The cluster size varies non-monotonically with activity: increasing first and decreasing at higher activity, thus driving cluster-to-fluid phase transition. Rotational diffusivity also facilitates the formation of clusters. Larger clusters could be stabilized at lowD_rvalues while at highD_rvalues, clusters are stable even at higher activities. The analysis of the static structure factor of the system confirms that rotational diffusivity delays the cluster-to-fluid transition driven by activity.

Intrinsic structure perspective for MIPS interfaces in two-dimensional systems of active Brownian particles

Suspensions of active Brownian particles (ABPs) undergo motility-induced phase separation (MIPS) over a wide range of mean density and activity strength, which implies the spontaneous aggregation of particles due to the persistence of their direction of motion, even in the absence of an explicit attraction. Both similarities and qualitative differences have been obtained when the MIPS is analysed in the same terms as a liquid-gas phase coexistence in an equilibrium attractive system. Negative values of the mechanical surface tension have been reported, from the total forces across the interface, while stable fluctuations of the interfacial line could be interpreted as a positive capillary surface tension; in equilibrium liquid surfaces, these two magnitudes are equal. We present here the analysis of 2D-ABP interfaces in terms of the intrinsic density and force profiles, calculated with the particle distance to the instantaneous interfacial line. Our results provide new insight into the origin of MIPS from the local rectification of the random active force on the particles near the interface. As has been reported, this effect acts as an external potential that produces a pressure gradient across the interface, such that the mechanical surface tension of the MIPS cannot be described as that of equilibrium coexisting phases; however, our analysis shows that most of that effect comes from the tightly caged particles at the dense (inner) side of the MIPS interface, rather than from the free moving particles at the outer side that collide with the dense cluster. Moreover, a clear correlation appears between the decay of the hexatic order parameter at the dense slab and the end of the MIPS as the strength of the active force is lowered. We show that, using the strong active forces required for MIPS, the interfacial structure and properties are very similar for ABPs with purely repulsive interactions (the Weeks-Chandler-Andersen-Lennard-Jones (WCA-LJ) model truncated at its minimum) and when the interaction includes a range of the LJ attractive force.

Extension of Kelvin's equation to dipolar colloids

Suspensions of colloids driven out-of-equilibrium demonstrate interesting collective behavior, such as organized and directed clustering and swarming. These systems require continuous energy input, yet some of the dynamics of these driven systems resemble the equilibrium-phase behavior of molecular fluids, such as crystallization, condensation, and phase separation. Consequently, there has been significant interest in exploring the applicability of thermodynamic concepts, such as pressure and surface tension, to describe nonequilibrium phenomena. Here, we show how rotating magnetic fields can drive superparamagnetic particles to form steady-state vapor–liquid coexistence that can be analyzed with Kelvin’s equation to determine an “effective vapor pressure” for this active colloidal system. These results illustrate the convergence of statistical physics of simple liquids to nonequilibrium colloidal fluids.

Vapor pressure refers to the pressure exerted by the vapor phase in thermodynamic equilibrium with either its liquid or solid phase. An important class of active matter is field-driven colloids. A suspension of dipolar colloids placed in a high-frequency rotating magnetic field undergoes a nonequilibrium phase transition into a dilute and dense phase, akin to liquid–vapor coexistence in a simple fluid. Here, we compute the vapor pressure of this colloidal fluid. The number of particles that exist as the dilute bulk phase versus condensed cluster phases can be directly visualized. An exponential relationship between vapor pressure and effective temperature is determined as a function of applied field strength, analogous to the thermodynamic expression between vapor pressure and temperature found for pure liquids. Additionally, we demonstrate the applicability of Kelvin’s equation to this field-driven system. In principle, this appears to be in conflict with macroscopic thermodynamic assumptions due to the nonequilibrium and discrete nature of this colloidal system. However, the curvature of the vapor–liquid interface provides a mechanical equilibrium characterized by interfacial tension that connects the condensed clusters observed with these active fluids to classical colligative fluid properties.

Adaptive Brownian Dynamics

A framework for performant Brownian Dynamics (BD) many-body simulations with adaptive timestepping is presented. Contrary to the Euler-Maruyama scheme in common non-adaptive BD, we employ an embedded Heun-Euler integrator for the propagation of the overdamped coupled Langevin equations of motion. This enables the derivation of a local error estimate and the formulation of criteria for the acceptance or rejection of trial steps and for the control of optimal stepsize. Introducing erroneous bias in the random forces is avoided by rejection sampling with memory due to Rackauckas and Nie, which makes use of the Brownian bridge theorem and guarantees the correct generation of a specified random process even when rejecting trial steps. For test cases of Lennard-Jones fluids in bulk and in confinement, it is shown that adaptive BD solves performance and stability issues of conventional BD, already outperforming the latter even in standard situations. We expect this novel computational approach to BD to be especially helpful in long-time simulations of complex systems, e.g., in non-equilibrium, where concurrent slow and fast processes occur.

Phase behavior and surface tension of soft active Brownian particles

We study quasi two-dimensional, monodisperse systems of active Brownian particles (ABPs) for a range of activities, stiffnesses, and densities. We develop a microscopic, analytical method for predicting the dense phase structure formed after motility-induced phase separation (MIPS) has occurred, including the dense cluster's area fraction, interparticle pressure, and radius. Our predictions are in good agreement with our Brownian dynamics simulations. We, then, derive a continuum model to investigate the relationship between the predicted interparticle pressure, the swim pressure, and the macroscopic pressure in the momentum equation. We find that formulating the point-wise macroscopic pressure as the interparticle pressure and modeling the particle activity through a spatially variant body force - as opposed to a volume-averaged swim pressure - results in consistent predictions of pressure in both the continuum model and the microscopic theory. This formulation of pressure also results in nearly zero surface tension for the phase separated domains, irrespective of activity, stiffness, and area fraction. Furthermore, using Brownian dynamics simulations and our continuum model, we showed that both the interface width and surface tension, are intrinsic characteristics of the system. On the other hand, if we were to exclude the body force induced by activity, we find that the resulting surface tension values are linearly dependent on the size of the simulation, in contrast to the statistical mechanical definition of surface tension.

Polarization-density patterns of active particles in motility gradients

The colocalization of density modulations and particle polarization is a characteristic emergent feature of motile active matter in activity gradients. We employ the active-Brownian-particle model to derive precise analytical expressions for the density and polarization profiles of a single Janus-type swimmer in the vicinity of an abrupt activity step. Our analysis allows for an optional (but not necessary) orientation-dependent propulsion speed, as often employed in force-free particle steering. The results agree well with measurement data for a thermophoretic microswimmer presented in the companion paper [Söker et al., Phys. Rev. Lett. 126, 228001 (2021)10.1103/PhysRevLett.126.228001], and they can serve as a template for more complex applications, e.g., to motility-induced phase separation or studies of physical boundaries. The essential physics behind our formal results is robustly captured and elucidated by a schematic two-species "run-and-tumble" model.

Wetting of a solid surface by active matter

A lattice model is used to study repulsive active particles at a planar surface. A rejection-free Kinetic Monte Carlo method is employed to characterize the wetting behaviour. The model predicts a motility-induced phase separation of active particles, and the bulk coexistence of dense liquid-like and dilute vapour-like steady states is determined. An "ensemble", with a varying number of particles, analogous to a grand canonical ensemble in equilibrium, is introduced. The formation and growth of the liquid film between the solid surface and the vapour phase is investigated. At constant activity, as the system is brought towards coexistence from the vapour side, the thickness of the adsorbed film exhibits a divergent behaviour regardless of the activity. This suggests a complete wetting scenario along the full coexistence curve.

The "isothermal" compressibility of active matter

We demonstrate that the mechanically defined "isothermal" compressibility behaves as a thermodynamic-like response function for suspensions of active Brownian particles. The compressibility computed from the active pressure-a combination of the collision and unique swim pressures-is capable of predicting the critical point for motility induced phase separation, as expected from the mechanical stability criterion. We relate this mechanical definition to the static structure factor via an active form of the thermodynamic compressibility equation and find the two to be equivalent, as would be the case for equilibrium systems. This equivalence indicates that compressibility behaves like a thermodynamic response function, even when activity is large. Finally, we discuss the importance of the phase interface when defining an active chemical potential. Previous definitions of the active chemical potential are shown to be accurate above the critical point but breakdown in the coexistence region. Inclusion of the swim pressure in the mechanical compressibility definition suggests that the interface is essential for determining phase behavior.

A particle-field approach bridges phase separation and collective motion in active matter

Whereas self-propelled hard discs undergo motility-induced phase separation, self-propelled rods exhibit a variety of nonequilibrium phenomena, including clustering, collective motion, and spatio-temporal chaos. In this work, we present a theoretical framework representing active particles by continuum fields. This concept combines the simplicity of alignment-based models, enabling analytical studies, and realistic models that incorporate the shape of self-propelled objects explicitly. By varying particle shape from circular to ellipsoidal, we show how nonequilibrium stresses acting among self-propelled rods destabilize motility-induced phase separation and facilitate orientational ordering, thereby connecting the realms of scalar and vectorial active matter. Though the interaction potential is strictly apolar, both, polar and nematic order may emerge and even coexist. Accordingly, the symmetry of ordered states is a dynamical property in active matter. The presented framework may represent various systems including bacterial colonies, cytoskeletal extracts, or shaken granular media.

Interacting self-propelled particles exhibit phase separation or collective motion depending on particle shape. A unified theory connecting these paradigms represents a major challenge in active matter, which the authors address here by modeling active particles as continuum fields.

Aggregate morphology of active Brownian particles on porous, circular walls

We study the motility-induced aggregation of active Brownian particles (ABPs) on a porous, circular wall. We observe that the morphology of aggregated dense-phase on a static wall depends on the wall porosity, particle motility, and the radius of the circular wall. Our analysis reveals two morphologically distinct, dense aggregates; a connected dense cluster that spreads uniformly on the circular wall and a localized cluster that breaks the rotational symmetry of the system. These distinct morphological states are similar to the macroscopic structures observed in aggregates on planar, porous walls. We systematically analyze the parameter regimes where the different morphological states are observed. We further extend our analysis to motile circular rings. We show that the motile ring propels almost ballistically due to the force applied by the active particles when they form a localized cluster, whereas it moves diffusively when the active particles form a continuous cluster. This property demonstrates the possibility of extracting useful work from a system of ABPs, even without artificially breaking the rotational symmetry.

Morphological transitions of active Brownian particle aggregates on porous walls

Motility-induced wall aggregation of Active Brownian Particles (ABPs) is a well-studied phenomenon. Here, we study the aggregation of ABPs on porous walls, which allows the particles to penetrate through at large motility. We show that the active aggregates undergo a morphological transition from a connected dense-phase to disconnected droplets with an increase in wall porosity and the particle self-motility, similar to wetting-dewetting transitions in equilibrium fluids. We show that both morphologically distinct states are stable, and independent of initial conditions at least in some parameter regions. Our analysis reveals that changes in wall porosity affect the intrinsic properties of the aggregates and changes the effective wall-aggregate interfacial tension, consistent with the appearance of the morphological transition. Accordingly, a close analysis of the density, as well as orientational distribution, indicates that the underlying reason for such morphological transitions is not necessarily specific to the systems with porous walls, and it can be possible to observe in a larger class of confined, active systems by tuning the properties of confining walls.

Surface Tensions between Active Fluids and Solid Interfaces: Bare vs Dressed

We analyze the surface tension exerted at the interface between an active fluid and a solid boundary in terms of tangential forces. Focusing on active systems known to possess an equation of state for the pressure, we show that interfacial forces are of a more complex nature. Using a number of macroscopic setups, we show that the surface tension is a combination of an equation-of-state abiding part and of setup-dependent contributions. The latter arise from generic setup-dependent steady currents which "dress" the measurement of the "bare" surface tension. The former shares interesting properties with its equilibrium counterpart, and can be used to generalize the Young-Laplace law to active systems. Finally, we show how a suitably designed probe can directly access this bare surface tension, which can also be computed using a generalized virial formula.

Collective forces in scalar active matter

Large-scale collective behavior in suspensions of active particles can be understood from the balance of statistical forces emerging beyond the direct microscopic particle interactions. Here we review some aspects of the collective forces that can arise in suspensions of self-propelled active Brownian particles: wall forces under confinement, interfacial forces, and forces on immersed bodies mediated by the suspension. Even for non-aligning active particles, these forces are intimately related to a non-uniform polarization of particle orientations induced by walls and bodies, or inhomogeneous density profiles. We conclude by pointing out future directions and promising areas for the application of collective forces in synthetic active matter, as well as their role in living active matter.

Microscopic origins of the swim pressure and the anomalous surface tension of active matter

The unique pressure exerted by active particles-the "swim" pressure-has proven to be a useful quantity in explaining many of the seemingly confounding behaviors of active particles. However, its use has also resulted in some puzzling findings including an extremely negative surface tension between phase separated active particles. Here, we demonstrate that this contradiction stems from the fact that the swim pressure is not a true pressure. At a boundary or interface, the reduction in particle swimming generates a net active force density-an entirely self-generated body force. The pressure at the boundary, which was previously identified as the swim pressure, is in fact an elevated (relative to the bulk) value of the traditional particle pressure that is generated by this interfacial force density. Recognizing this unique mechanism for stress generation allows us to define a much more physically plausible surface tension. We clarify the utility of the swim pressure as an "equivalent pressure" (analogous to those defined from electrostatic and gravitational body forces) and the conditions in which this concept can be appropriately applied.

Non-negative Interfacial Tension in Phase-Separated Active Brownian Particles

We present a microscopic theory for the nonequilibrium interfacial tension γ_{gl} of the free interface between gas and liquid phases of active Brownian particles. The underlying square gradient treatment and the splitting of the force balance in flow and structural contributions is general and applies to inhomogeneous nonequilibrium steady states. We find γ_{gl}≥0, which opposes claims by Bialké et al. [Phys. Rev. Lett. 115, 098301 (2015)PRLTAO0031-900710.1103/PhysRevLett.115.098301] and delivers the theoretical justification for the widely observed interfacial stability in active Brownian dynamics many-body simulations.

Phase coexistence of active Brownian particles

We investigate motility-induced phase separation of active Brownian particles, which are modeled as purely repulsive spheres that move due to a constant swim force with freely diffusing orientation. We develop on the basis of power functional concepts an analytical theory for nonequilibrium phase coexistence and interfacial structure. Theoretical predictions are validated against Brownian dynamics computer simulations. We show that the internal one-body force field has four nonequilibrium contributions: (i) isotropic drag and (ii) interfacial drag forces against the forward motion, (iii) a superadiabatic spherical pressure gradient, and (iv) the quiet life gradient force. The intrinsic spherical pressure is balanced by the swim pressure, which arises from the polarization of the free interface. The quiet life force opposes the adiabatic force, which is due to the inhomogeneous density distribution. The balance of quiet life and adiabatic forces determines bulk coexistence via equality of two bulk state functions, which are independent of interfacial contributions. The internal force fields are kinematic functionals which depend on density and current but are independent of external and swim forces, consistent with power functional theory. The phase transition originates from nonequilibrium repulsion, with the agile gas being more repulsive than the quiet liquid.

Curvature-dependent tension and tangential flows at the interface of motility-induced phases

Purely repulsive active particles spontaneously undergo motility-induced phase separation (MIPS) into condensed and dilute phases. Remarkably, the mechanical tension measured along the interface between these phases is negative. In equilibrium this would imply an unstable interface that wants to expand, but these out-of-equilibrium systems display long-time stability and have intrinsically stiff boundaries. Here, we study this phenomenon in detail using active Brownian particle simulations and a novel frame of reference. By shifting from the global (or laboratory) frame to a local frame that follows the dynamics of the phase boundary, we observe correlations between the local curvature of the interface and the measured value of the tension. Importantly, our analysis reveals that curvature drives sustained local tangential motion of particles within a surface layer in both the gas and the dense regions. The combined tangential current in the gas and local "self-shearing" of the surface of the dense phase suggest a stiffening interface that redirects particles along itself to heal local fluctuations. These currents restore the otherwise wildly fluctuating interface through an out-of-equilibrium Marangoni effect. We discuss the implications of our observations on phenomenological models of interfacial dynamics.

Interfacial Properties of Active-Passive Polymer Mixtures

Active matter consists of particles that dissipate energy, from their own sources, in the form of mechanical work on their surroundings. Recent interest in active-passive polymer mixtures has been driven by their relevance in phase separation of (e.g., transcriptionally) active and inactive (transcriptionally silent) DNA strands in nuclei of living cells. In this paper, we study the interfacial properties of the phase separated steady states of the active-passive polymer mixtures and compare them with equilibrium phase separation. We model the active constituents by assigning them stronger-than-thermal fluctuations. We demonstrate that the entropy production is an accurate indicator of the phase transition. We then construct phase diagrams and analyze kinetic properties of the particles as a function of the distance from the interface. Studying the interface fluctuations, we find that they follow the capillary waves spectrum. This allows us to establish a mechanistic definition of the interfacial stiffness and its dependence on the relative level of activity with respect to the passive constituents. We show how the interfacial width depends on the activity ratio and comment on the finite size effects. Our results highlight similarities and differences of the non-equilibrium steady states with an equilibrium phase separated polymer mixture with a lower critical solution temperature. We present several directions in which the non-equilibrium system can be studied further and point out interesting observations that indicate general principles behind the non-equilibrium phase separation.