Swim Pressure Research Articles

Exact moments and re-entrant transitions in the inertial dynamics of active Brownian particles

In this study, we investigate the behavior of free inertial active Brownian particles in the presence of thermal noise. While finding a closed-form solution for the joint distribution of positions, orientations, and velocities using the Fokker–Planck equation is generally challenging, we utilize a Laplace transform method to obtain the exact temporal evolution of all dynamical moments in arbitrary dimensions. Our expressions in d dimensions reveal that inertia significantly impacts steady-state kinetic temperature and swim pressure while leaving the late-time diffusivity unchanged. Notably, as a function of activity and inertia, the steady-state velocity distribution exhibits a remarkable re-entrant crossover from ‘passive’ Gaussian to ‘active’ non-Gaussian behaviors. We construct a corresponding ‘phase diagram’ using the exact expression of the d-dimensional kurtosis. Our analytic expressions describe steady states and offer insights into time-dependent crossovers observed in moments of velocity and displacement. Our calculations can be extended to predict up to second-order moments for run-and-tumble particles and the active Ornstein–Uhlenbeck process (AOUP). Additionally, the kurtosis shows differences from AOUP.

Density and polarization of active Brownian particles in curved activity landscapes.

Suspensions of motile active particles with space-dependent activity form characteristic polarization and density patterns. Recent single-particle studies for planar activity landscapes identified several quantities associated with emergent density-polarization patterns that are solely determined by bulk variables. Naive thermodynamic intuition suggests that these results might hold for arbitrary activity landscapes mediating bulk regions, and thus could be used as benchmarks for simulations and theories. However, the considered system operates in a nonequilibrium steady state and we prove by construction that the quantities in question lose their simple form for curved activity landscapes. Specifically, we provide a detailed analytical study of polarization and density profiles induced by radially symmetric activity steps, and of the total polarization for the case of a general radially symmetric activity landscape. While the qualitative picture is similar to the planar case, all the investigated variables depend not only on bulk variables but also comprise geometry-induced contributions. We verified that all our analytical results agree with exact numerical calculations.

Active particles with polar alignment in ring-shaped confinement.

We study steady-state properties of active, nonchiral and chiral Brownian particles with polar alignment and steric interactions confined within a ring-shaped confinement (annulus) in two dimensions. Exploring possible interplays between polar interparticle alignment, geometric confinement and the surface curvature, being incorporated here on minimal levels, we report a surface-population reversal effect, whereby active particles migrate from the outer concave boundary of the annulus to accumulate on its inner convex boundary. This contrasts the conventional picture, implying stronger accumulation of active particles on concave boundaries relative to the convex ones. The population reversal is caused by both particle alignment and surface curvature, disappearing when either of these factors is absent. We explore the ensuing consequences for the chirality-induced current and swim pressure of active particles and analyze possible roles of system parameters, such as the mean number density of particles and particle self-propulsion, chirality, and alignment strengths.

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.

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.

Inertial effects on trapped active matter.

In this work, the dynamics of inertial (mass and moment of inertia) active Brownian particles trapped in a harmonic well is studied. This scenario has seen success when characterizing soft passive and active overdamped matter. Motivated by the variety of applications of this system, we analytically find the effect of translational and rotational inertia on the mean-square displacement (MSD), mean-square speed (MSS), swim, Reynolds, and total pressures of a system of inertial active Brownian particles subject to a weak and a strong harmonic trap. Following a Langevin formalism, we explicitly find that as inertia grows, the systems' MSD and total pressure are enhanced, but its MSS and swim pressure decrease. The use of Langevin dynamics simulations enables us to observe that as inertia grows, inertial active matter under a strong trap no longer "condensates" at the "border" of the trap, but it rather tends to uniformly spread in space. Our analytical results are also numerically validated.

Testing a thermodynamic approach to collective animal behavior in laboratory fish schools

Collective behaviors displayed by groups of social animals are observed frequently in nature. Understanding and predicting the behavior of complex biological systems is dependent on developing effective descriptions and models. While collective animal systems are characteristically nonequilibrium, we can employ concepts from equilibrium statistical mechanics to motivate the measurement of material-like properties in laboratory animal aggregates. Here, we present results from a new set of experiments that utilize high speed footage of two-dimensional schooling events, particle tracking, and projected static and dynamic light fields to observe and control the behavior of negatively phototaxic fish schools (Hemigrammus bleheri). First, we use static light fields consisting of dark circular regions to produce visual stimuli that confine the schools to a range of areas. We find that schools have a maximum density which is independent of group size, and that a swim pressurelike quantity, Π increases linearly with number density, suggesting that unperturbed schools exist on an isotherm. Next, we use dynamic light fields where the radius of the dark region shrinks linearly with time to compress the schools. We find that an effective temperature parameter depends on the compression time and our results are thus consistent with the school having a constant heat flux. These findings further evidence the utility of effective thermodynamic descriptions of nonequilibrium systems in collective animal behavior.

Compaction Dynamics during Progenitor Cell Self-Assembly Reveal Granular Mechanics

Summary We study the self-assembly dynamics of human progenitor cells in agarose microwells that are used for production of chondrogenic organoids. Using image analysis on time-lapse microscopy, we estimate the aggregate area in function of time for a large number of aggregates. In control conditions, the aggregate radius follows an exponential relaxation that is consistent with the dewetting dynamics of a liquid film. Upon introducing Y-27632 Rho kinase inhibitor, the compatibility with the liquid model is lost, and slowed-down relaxation dynamics are observed. We demonstrate that these aggregates behave as granular piles undergoing compaction, with density relaxation that follows a stretched exponential. Using simulations with an individual cell-based model, we construct a phase diagram of cell aggregates that suggests that the aggregate in presence of Rho kinase inhibitor approaches glass transition.

Pressure and diffusion of active matter with inertia.

It has been discovered that active matter generates novel physical quantities such as the swim pressure. This quantity arises from the exchange of extra momentum between active particles and the boundaries of the system. Given its origin, this quantity can exist at different scales; hence microorganisms and larger organisms like fish or birds generate their own swim pressure. For larger organisms or for high swimming speeds, inertia cannot necessarily be neglected; hence in this paper, we start by calculating analytically the effect of finite translational and rotational particles' inertia on the diffusion of a system of noninteracting spherical active Brownian particles. From this analysis, an enhanced diffusion coefficient due to rotational inertia is obtained, and an alternative effective persistence length and an alternative reorientation time, both sensitive to rotational inertia, are also identified. Afterwards, and to see the implications of finite inertia on bulk properties, the pressure of this system is elucidated by calculating its respective swim and Reynolds pressures. It is found that their sum becomes asymptotically sensitive to the square root of its rotational inertia. To validate our analytical results, Langevin dynamics simulations are also performed showing an excellent agreement between our theoretical predictions and the numerical results.

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.

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.

Shape Transformations of Vesicles Induced by Swim Pressure.

While the behavior of vesicles in thermodynamic equilibrium has been studied extensively, how active forces control vesicle shape transformations is not understood. Here, we combine theory and simulations to study the shape behavior of vesicles containing active Brownian particles. We show that the combination of active forces, dimensionality, and membrane bending free energy creates a plethora of novel phase transitions. At low swim pressure, the vesicle exhibits a discontinuous transition from a spherical to a prolate shape, which has no counterpart in two dimensions. At high swim pressure it exhibits stochastic spatiotemporal oscillations. Our work helps researchers to understand and control the shape dynamics of membranes in active-matter systems.

Statistical mechanics of transport processes in active fluids. II. Equations of hydrodynamics for active Brownian particles.

We perform a coarse-graining analysis of the paradigmatic active matter model, active Brownian particles, yielding a continuum description in terms of balance laws for mass, linear and angular momentum, and energy. The derivation of the balance of linear momentum reveals that the active force manifests itself directly as a continuum-level body force proportional to an order parameter-like director field, which therefore requires its own evolution equation to complete the continuum description of the system. We derive this equation, demonstrating in the process that bulk currents may be sustained in homogeneous systems only in the presence of interparticle aligning interactions. Furthermore, we perform a second coarse-graining of the balance of linear momentum and derive the expression for active or swim pressure in the case of mechanical equilibrium.

Anisotropic swim stress in active matter with nematic order

Active Brownian particles (ABPs) transmit a swim pressure to the container boundaries, where ζ is the drag coefficient, Dswim is the swim diffusivity and n is the uniform bulk number density far from the container walls. In this work we extend the notion of the isotropic swim pressure to the anisotropic tensorial swim stress , which is related to the anisotropic swim diffusivity . We demonstrate this relationship with ABPs that achieve nematic orientational order via a bulk external field. The anisotropic swim stress is obtained analytically for dilute ABPs in both 2D and 3D systems. The anisotropy, defined as the ratio of the maximum to the minimum of the three principal stresses, is shown to grow exponentially with the strength of the external field. We verify that the normal component of the anisotropic swim stress applies a pressure on a wall with normal vector , and, through Brownian dynamics simulations, this pressure is shown to be the force per unit area transmitted by the active particles. Since ABPs have no friction with a wall, the difference between the normal and tangential stress components—the normal stress difference—generates a net flow of ABPs along the wall, which is a generic property of active matter systems.

Instability of expanding bacterial droplets

Suspensions of motile bacteria or synthetic microswimmers, termed active matter, manifest a remarkable propensity for self-organization, and formation of large-scale coherent structures. Most active matter research deals with almost homogeneous in space systems and little is known about the dynamics of strongly heterogeneous active matter. Here we report on experimental and theoretical studies on the expansion of highly concentrated bacterial droplets into an ambient bacteria-free fluid. The droplet is formed beneath a rapidly rotating solid macroscopic particle inserted in the suspension. We observe vigorous instability of the droplet reminiscent of a violent explosion. The phenomenon is explained in terms of continuum first-principle theory based on the swim pressure concept. Our findings provide insights into the dynamics of active matter with strong density gradients and significantly expand the scope of experimental and analytic tools for control and manipulation of active systems.

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.

The curved kinetic boundary layer of active matter.

A body submerged in active matter feels the swim pressure through a kinetic accumulation boundary layer on its surface. The boundary layer results from a balance between translational diffusion and advective swimming and occurs on the microscopic length scale . Here , DT is the Brownian translational diffusivity, τR is the reorientation time and l = U0τR is the swimmer's run length, with U0 the swim speed [Yan and Brady, J. Fluid. Mech., 2015, 785, R1]. In this work we analyze the swim pressure on arbitrary shaped bodies by including the effect of local shape curvature in the kinetic boundary layer. When δ ≪ L and l ≪ L, where L is the body size, the leading order effects of curvature on the swim pressure are found analytically to scale as JSλδ2/L, where JS is twice the (non-dimensional) mean curvature. Particle-tracking simulations and direct solutions to the Smoluchowski equation governing the probability distribution of the active particles show that λδ2/L is a universal scaling parameter not limited to the regime δ, l ≪ L. The net force exerted on the body by the swimmers is found to scale as Fnet/(n∞ksTsL2) = f(λδ2/L), where f(x) is a dimensionless function that is quadratic when x ≪ 1 and linear when x ∼ 1. Here, ksTs= ζU02τR/6 defines the 'activity' of the swimmers, with ζ the drag coefficient, and n∞ is the uniform number density of swimmers far from the body. We discuss the connection of this boundary layer to continuum mechanical descriptions of active matter and briefly present how to include hydrodynamics into this purely kinetic study.

Do hydrodynamic interactions affect the swim pressure?

We study the motion of a spherical active Brownian particle (ABP) of size a, moving with a fixed speed U0, and reorienting on a time scale τR in the presence of a confining boundary. Because momentum is conserved in the embedding fluid, we show that the average force per unit area on the boundary equals the bulk mechanical pressure P∞ = p∞f + Π∞, where p∞f is the fluid pressure and Π∞ is the particle pressure; this is true for active and passive particles alike regardless of how the particles interact with the boundary. As an example, we investigate how hydrodynamic interactions (HI) change the particle-phase pressure at the wall, and find that Πwall = n∞(kBT + ζ(Δ)U0l(Δ)/6), where ζ is the (Stokes) drag on the swimmer, l = U0τR is the run length, and Δ is the minimum gap size between the particle and the wall; as Δ → ∞ this is the familiar swim pressure [Takatori et al., Phys. Rev. Lett., 2014, 113, 1-5].

Kinetics of motility-induced phase separation and swim pressure.

Active Brownian particles (ABPs) represent a minimal model of active matter consisting of self-propelled spheres with purely repulsive interactions and rotational noise. Here we examine the pressure of ABPs in two dimensions in both closed boxes and systems with periodic boundary conditions and show that its nonmonotonic behavior with density is a general property of ABPs and is not the result of finite-size effects. We correlate the time evolution of the mean pressure towards its steady-state value with the kinetics of motility-induced phase separation. For parameter values corresponding to phase-separated steady states, we identify two dynamical regimes. The pressure grows monotonically in time during the initial regime of rapid cluster formation, overshooting its steady-state value and then quickly relaxing to it, and remains constant during the subsequent slower period of cluster coalescence and coarsening. The overshoot is a distinctive feature of active systems.

Acoustic trapping of active matter.

Confinement of living microorganisms and self-propelled particles by an external trap provides a means of analysing the motion and behaviour of active systems. Developing a tweezer with a trapping radius large compared with the swimmers' size and run length has been an experimental challenge, as standard optical traps are too weak. Here we report the novel use of an acoustic tweezer to confine self-propelled particles in two dimensions over distances large compared with the swimmers' run length. We develop a near-harmonic trap to demonstrate the crossover from weak confinement, where the probability density is Boltzmann-like, to strong confinement, where the density is peaked along the perimeter. At high concentrations the swimmers crystallize into a close-packed structure, which subsequently ‘explodes' as a travelling wave when the tweezer is turned off. The swimmers' confined motion provides a measurement of the swim pressure, a unique mechanical pressure exerted by self-propelled bodies.