Interface stability, interface fluctuations, and the Gibbs-Thomson relationship in motility-induced phase separations

Interfacial and density fluctuations in a lattice model of motility-induced phase separation

We analyze motility-induced phase separation and bubbly phase separation in a two-dimensional lattice model of self-propelled particles. We compare systems where the dense (liquid) phase has slab and droplet geometries. We find that interfacial fluctuations of the slab are well-described by capillary wave theory, despite the existence of bubbles in the dense phase. We attribute this to a separation of time scales between bubble expulsion and interfacial relaxation. We also characterize the dependence of liquid and vapor densities on the curvature of the liquid droplet, as well as the density fluctuations inside the phases. The vapor phase behaves similarly to an equilibrium system, displaying a Laplace pressure effect that shifts its density, and Gaussian density fluctuations. The liquid phase has large non-Gaussian fluctuations, but this is not accompanied by a large density shift, contrary to the equilibrium case. Nevertheless, the shift of the vapor density can be used to infer an effective surface tension that appears to also quantify capillary wave fluctuations.

Phase Separation, Capillarity, and Odd-Surface Flows in Chiral Active Matter

Active phase separations evade canonical thermodynamic descriptions and have thus challenged our understanding of coexistence and interfacial phenomena. Considerable progress has been made towards a nonequilibrium theoretical description of these traditionally thermodynamic concepts. Spatial parity symmetry is conspicuously assumed in much of this progress, despite the ubiquity of chirality in experimentally realized systems. In this Letter, we derive a theory for the phase coexistence and interfacial fluctuations of a system that microscopically violates spatial parity. We find suppression of the phase separation as chirality is increased as well as the development of steady-state currents tangential to the interface dividing the phases. These odd flows are irrelevant to stationary interfacial properties, with stability, capillary fluctuations, and surface area minimization determined entirely by the capillary surface tension. Using large-scale Brownian dynamics simulations, we find excellent agreement with our theoretical scaling predictions.

Theory of capillary tension and interfacial dynamics of motility-induced phases

The statistical mechanics of equilibrium interfaces has been well-established for over a half century. In the past decade, a wealth of observations have made increasingly clear that a new perspective is required to describe interfaces arbitrarily far from equilibrium. In this work, beginning from microscopic particle dynamics that break time-reversal symmetry, we derive the linear interfacial dynamics of coexisting motility-induced phases. Doing so allows us to identify the athermal energy scale that excites interfacial fluctuations and the nonequilibrium surface tension that resists these excitations. Our theory identifies that, in contrast to equilibrium fluids, this active surface tension contains contributions arising from nonconservative forces which act to suppress interfacial fluctuations and, crucially, is distinct from the mechanical surface tension of Kirkwood and Buff. We find that the interfacial stiffness scales linearly with the intrinsic persistence length of the constituent active particle trajectories, in agreement with simulation data. We demonstrate that at wavelengths much larger than the persistence length, the interface obeys surface-area minimizing Boltzmann statistics with our derived nonequilibrium interfacial stiffness playing a role identical to that of equilibrium systems.

Statistical properties of microphase and bubbly phase-separated active fluids

In phase-separated active fluids, the Ostwald process can go into reverse, leading to either microphase separation or bubbly phase separation. We show that the latter is formed of two macroscopic regions that are occupied by the homogeneous fluid and by the microphase separated one. Within the microphase-separated fluid, the relative rate of the Ostwald process, coalescence, and nucleation determines whether the size distribution of mesoscopic domains is narrowly peaked or displays a broad range of sizes before attaining a cutoff independent of system size. Our results are obtained via large-scale simulations of a minimal field theory for active phase separation and reproduced by an effective model in which the degrees of freedom are the locations and sizes of the microphase-separated domains.

Activity affects the stability, deformation and breakage dynamics of colloidal architectures

Living network architectures, such as the cytoskeleton, are characterized by continuous energy injection, leading to rich but poorly understood non-equilibrium physics. There is a need for a well-controlled (experimental) model system that allows basic insight into such non-equilibrium processes. Activated self-assembled colloidal architectures can fulfill this role, as colloidal patchy particles can self-assemble into colloidal architectures such as chains, rings and networks, while self-propelled colloidal particles can simultaneously inject energy into the architecture, alter the dynamical behavior of the system, and cause the self-assembled structures to deform and break. To gain insight, we conduct a numerical investigation into the effect of introducing self-propelled colloids modeled as active Brownian particles, into self-assembling colloidal dispersions of dipatch and tripatch particles. For the interaction potential, we use a previously designed model that accurately can reproduce experimental colloidal self-assembly via the critical Casimir force [Jonas et al., J. Chem. Phys., 2021, 135, 034902]. Here, we focus primarily on the breakage dynamics of three archetypal substructures, namely, dimers, chains, and rings. We find a rich response behavior to the introduction of self-propelled particles, in which the activity can enhance as well as reduce the stability of the architecture, deform the intact structures and alter the mechanisms of fragmentation. We rationalize these findings in terms of the rate and mechanisms of breakage as a function of the direction and magnitude of the active force by separating the bond breakage process into two stages: escaping the potential well and separation of the particles. The results set the stage for investigating more complex architectures.

Active Brownian particles in external force fields: Field-theoretical models, generalized barometric law, and programmable density patterns

We investigate the influence of external forces on the collective dynamics of interacting active Brownian particles in two as well as three spatial dimensions. Via explicit coarse graining, we derive predictive models, i.e., models that give a direct relation between the models' coefficients and the bare parameters of the system, that are applicable for space- and time-dependent external force fields. We study these models for the cases of gravity and harmonic traps. In particular, we derive a generalized barometric formula for interacting active Brownian particles under gravity that is valid for low to high concentrations and activities of the particles. Furthermore, we show that one can use an external harmonic trap to induce motility-induced phase separation in systems that, without external fields, remain in a homogeneous state. This finding makes it possible to realize programmable density patterns in systems of active Brownian particles. Our analytic predictions are found to be in very good agreement with Brownian dynamics simulations.

Interface Roughening in Nonequilibrium Phase-Separated Systems

Interfaces of phase-separated systems roughen in time due to capillary waves. Because of fluxes in the bulk, their dynamics is nonlocal in real space and is not described by the Edwards-Wilkinson or Kardar-Parisi-Zhang (KPZ) equations, nor their conserved counterparts. We show that, in the absence of detailed balance, the phase-separated interface is described by a new universality class that we term |q|KPZ. We compute the associated scaling exponents via one-loop renormalization group and corroborate the results by numerical integration of the |q|KPZ equation. Deriving the effective interface dynamics from a minimal field theory of active phase separation, we finally argue that the |q|KPZ universality class generically describes liquid-vapor interfaces in two- and three-dimensional active systems.

Dynamic overlap concentration scale of active colloids

By introducing the notion of a dynamic overlap concentration scale, we identify additional universal features of the mechanical properties of active colloids. We codify these features by recognizing that the characteristic length scale of an active particle's trajectory, the run length, introduces a concentration scale ϕ^{*}. Large-scale simulations of repulsive active Brownian particles (ABPs) confirm that this run-length dependent concentration, the trajectory-space analog of the overlap concentration in polymer solutions, delineates distinct concentration regimes in which interparticle collisions alter particle trajectories. Using ϕ^{*} and concentration scales associated with colloidal jamming, the mechanical equation of state for ABPs collapses onto a set of principal curves that contain several overlooked features. The inclusion of these features qualitatively alters previous predictions of the behavior for active colloids, as we demonstrate by computing the spinodal for a suspension of purely repulsive ABPs. Our findings suggest that dynamic overlap concentration scales should help unravel the behavior of active and driven systems.

Capillary Interfacial Tension in Active Phase Separation

In passive fluid-fluid phase separation, a single interfacial tension sets both the capillary fluctuations of the interface and the rate of Ostwald ripening. We show that these phenomena are governed by two different tensions in active systems, and compute the capillary tension σ_{cw} which sets the relaxation rate of interfacial fluctuations in accordance with capillary wave theory. We discover that strong enough activity can cause negative σ_{cw}. In this regime, depending on the global composition, the system self-organizes, either into a microphase-separated state in which coalescence is highly inhibited, or into an "active foam" state. Our results are obtained for Active Model B+, a minimal continuum model which, although generic, admits significant analytical progress.

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.

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.

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.

Critical Motility-Induced Phase Separation Belongs to the Ising Universality Class

A collection of self-propelled particles with volume exclusion interactions can exhibit the phenomenology of a gas-liquid phase separation, known as motility-induced phase separation (MIPS). The nonequilibrium nature of the system is fundamental to the phase transition; however, it is unclear whether MIPS at criticality contributes a novel universality class to nonequilibrium physics. We demonstrate here that this is not the case by showing that a generic critical MIPS belongs to the Ising universality class with conservative dynamics.

Physical Chemistry of Cellular Liquid-Phase Separation

Compartmentalization of biochemical processes is essential for cell function. Although membrane-bound organelles are well studied in this context, recent work has shown that phase separation is a key contributor to cellular compartmentalization through the formation of liquid-like membraneless organelles (MLOs). In this Minireview, the key mechanistic concepts that underlie MLO dynamics and function are first briefly discussed, including the relevant noncovalent interaction chemistry and polymer physical chemistry. Next, a few examples of MLOs and relevant proteins are given, along with their functions, which highlight the relevance of the above concepts. The developing area of active matter and non-equilibrium systems, which can give rise to unexpected effects in fluctuating cellular conditions, are also discussed. Finally, our thoughts for emerging and future directions in the field are discussed, including in vitro and in vivo studies of MLO physical chemistry and function.

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.

Active fluids at circular boundaries: swim pressure and anomalous droplet ripening

We investigate the swim pressure exerted by non-chiral and chiral active particles on convex or concave circular boundaries. Active particles are modeled as non-interacting and non-aligning self-propelled Brownian particles. The convex and concave circular boundaries are used to model a fixed inclusion immersed in an active bath and a cavity (or container) enclosing the active particles, respectively. We first present a detailed analysis of the role of convex versus concave boundary curvature and of the chirality of active particles in their spatial distribution, chirality-induced currents, and the swim pressure they exert on the bounding surfaces. The results will then be used to predict the mechanical equilibria of suspended fluid enclosures (generically referred to as 'droplets') in a bulk with active particles being present either inside the bulk fluid or within the suspended droplets. We show that, while droplets containing active particles behave in accordance with standard capillary paradigms when suspended in a normal bulk, those containing a normal fluid exhibit anomalous behaviors when suspended in an active bulk. In the latter case, the excess swim pressure results in non-monotonic dependence of the inside droplet pressure on the droplet radius; hence, revealing an anomalous regime of behavior beyond a threshold radius, in which the inside droplet pressure increases upon increasing the droplet size. Furthermore, for two interconnected droplets, mechanical equilibrium can occur also when the droplets have different sizes. We thus identify a regime of anomalous droplet ripening, where two unequal-sized droplets can reach a final state of equal size upon interconnection, in stark contrast with the standard Ostwald ripening phenomenon, implying shrinkage of the smaller droplet in favor of the larger one.

Physical principles of intracellular organization via active and passive phase transitions

Exciting recent developments suggest that phase transitions represent an important and ubiquitous mechanism underlying intracellular organization. We describe key experimental findings in this area of study, as well as the application of classical theoretical approaches for quantitatively understanding these data. We also discuss the way in which equilibrium thermodynamic driving forces may interface with the fundamentally out-of-equilibrium nature of living cells. In particular, time and/or space-dependent concentration profiles may modulate the phase behavior of biomolecules in living cells. We suggest future directions for both theoretical and experimental work that will shed light on the way in which biological activity modulates the assembly, properties, and function of viscoelastic states of living matter.