Mechanical theory of nonequilibrium coexistence and motility-induced phase separation.

Nonequilibrium phase transitions are routinely observed in both natural and synthetic systems. The ubiquity of these transitions highlights the conspicuous absence of a general theory of phase coexistence that is broadly applicable to both nonequilibrium and equilibrium systems. Here, we present a general mechanical theory for phase separation rooted in ideas explored nearly a half-century ago in the study of inhomogeneous fluids. The core idea is that the mechanical forces within the interface separating two coexisting phases uniquely determine coexistence criteria, regardless of whether a system is in equilibrium or not. We demonstrate the power and utility of this theory by applying it to active Brownian particles, predicting a quantitative phase diagram for motility-induced phase separation in both two and three dimensions. This formulation additionally allows for the prediction of novel interfacial phenomena, such as an increasing interface width while moving deeper into the two-phase region, a uniquely nonequilibrium effect confirmed by computer simulations. The self-consistent determination of bulk phase behavior and interfacial phenomena offered by this mechanical perspective provide a concrete path forward toward a general theory for nonequilibrium phase transitions.

Confined active matter in external fields.

We analyze a dilute suspension of active particles confined between walls and subjected to fields that can modulate particle speed as well as orientation. Generally, the particle distribution is different in the bulk compared to near the walls. In the bulk, particles tend to accumulate in the regions of low speed, but in the presence of an orienting field normal to the walls, particles rotate to align with the field and accumulate in the field direction. At the walls, particles tend to accumulate pointing into the walls and thereby exert pressure on walls. But the presence of strong orienting fields can cause the particles to reorient away from the walls, and hence shows a possible mechanism for preventing contamination of surfaces. The pressure at the walls depends on the wall separation and the field strengths. This work demonstrates how multiple fields with different functionalities can be used to control active matter under confinement.

Tuning nonequilibrium phase transitions with inertia.

In striking contrast to equilibrium systems, inertia can profoundly alter the structure of active systems. Here, we demonstrate that driven systems can exhibit effective equilibrium-like states with increasing particle inertia, despite rigorously violating the fluctuation-dissipation theorem. Increasing inertia progressively eliminates motility-induced phase separation and restores equilibrium crystallization for active Brownian spheres. This effect appears to be general for a wide class of active systems, including those driven by deterministic time-dependent external fields, whose nonequilibrium patterns ultimately disappear with increasing inertia. The path to this effective equilibrium limit can be complex, with finite inertia sometimes acting to accentuate nonequilibrium transitions. The restoration of near equilibrium statistics can be understood through the conversion of active momentum sources to passive-like stresses. Unlike truly equilibrium systems, the effective temperature is now density dependent, the only remnant of the nonequilibrium dynamics. This density-dependent temperature can in principle introduce departures from equilibrium expectations, particularly in response to strong gradients. Our results provide additional insight into the effective temperature ansatz while revealing a mechanism to tune nonequilibrium phase transitions.

Partitioning of active particles into porous media.

Passive Brownian particles partition homogeneously between a porous medium and an adjacent fluid reservoir. In contrast, active particles accumulate near boundaries and can therefore preferentially partition into the porous medium. Understanding how active particles interact with and partition into such an environment is important for optimizing particle transport. In this work, both the initial transient and steady behavior as active swimmers partition into a porous medium from a bulk fluid reservoir are investigated. At short times, the particle number density in the porous medium exhibits an oscillatory behavior due to the particles' ballistic motion when time t < τR, where τR is the reorientation time of the active particles. At longer times, t > L2/Dswim, the particles diffuse from the reservoir into the porous medium, leading to a steady state concentration partitioning. Here, L is the characteristic length scale of the porous medium and Dswim = U0/d(d - 1), where U0 is the intrinsic swim speed of the particles,  = U0τR is the particles' run, or persistence, length, and d is the dimension of the reorientation process. An analytical prediction is developed for this partitioning for spherical obstacles connected to a fluid reservoir in both two and three dimensions based on the Smoluchowski equation and a macroscopic mechanical momentum balance. The analytical prediction agrees well with Brownian dynamics simulations.

Trapped-particle microrheology of active suspensions.

In microrheology, the local rheological properties, such as the viscoelasticity of a complex fluid, are inferred from the free or forced motion of embedded colloidal probe particles. Theoretical machinery developed for forced-probe microrheology of colloidal suspensions focused on either constant-force (CF) or constant-velocity (CV) probes, while in experiments, neither the force nor the kinematics of the probe is fixed. More importantly, the constraint of CF or CV introduces a difficulty in the meaningful quantification of the fluctuations of the probe due to a thermodynamic uncertainty relation. It is known that, for a Brownian particle trapped in a harmonic potential well, the product of the standard deviations of the trap force and the particle position is dkBT in d dimensions, with kBT being the thermal energy. As a result, if the force (position) is not allowed to fluctuate, the position (force) fluctuation becomes infinite. To allow the measurement of fluctuations in theoretical studies, in this work, we consider a microrheology model in which the embedded probe is dragged along by a moving harmonic potential so that both its position and the trap force are allowed to fluctuate. Starting from the full Smoluchowski equation governing the dynamics of N hard active Brownian particles, we derive a pair Smoluchowski equation describing the dynamics of the probe as it interacts with one bath particle by neglecting hydrodynamic interactions among particles in the dilute limit. From this, we determine the mean and the variance (i.e., fluctuation) of the probe position in terms of the pair probability distribution. We then characterize the behavior of the system in the limits of both weak and strong trap. By taking appropriate limits, we show that our generalized model can be reduced to the well-studied CF or CV microrheology models.

Machine learning for phase behavior in active matter systems.

We demonstrate that deep learning techniques can be used to predict motility-induced phase separation (MIPS) in suspensions of active Brownian particles (ABPs) by creating a notion of phase at the particle level. Using a fully connected network in conjunction with a graph neural network we use individual particle features to predict to which phase a particle belongs. From this, we are able to compute the fraction of dilute particles to determine if the system is in the homogeneous dilute, dense, or coexistence region. Our predictions are compared against the MIPS binodal computed from simulation. The strong agreement between the two suggests that machine learning provides an effective way to determine the phase behavior of ABPs and could prove useful for determining more complex phase diagrams.

Theory for the Casimir effect and the partitioning of active matter.

Active Brownian particles (ABPs) distribute non-homogeneously near surfaces, and understanding how this depends on system properties-size, shape, activity level, etc.-is essential for predicting and exploiting the behavior of active matter systems. Active particles accumulate at no-flux surfaces owing to their persistent swimming, which depends on their intrinsic swim speed and reorientation time, and are subject to confinement effects when their run or persistence length is comparable to the characteristic size of the confining geometry. It has been observed in simulations that two parallel plates experience a "Casimir effect" and attract each other when placed in a dilute bath of ABPs. In this work, we provide a theoretical model based on the Smoluchowski equation and a macroscopic mechanical momentum balance to analytically predict this attractive force. We extend this method to describe the concentration partitioning of active particles between a confining channel and a reservoir, showing that the ratio of the concentration in the channel to that in the bulk increases as either run length increases or channel height decreases. The theoretical results agree well with Brownian dynamics simulations and finite element calculations.

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.

Nonlinear microrheology of active Brownian suspensions.

The rheological properties of active suspensions are studied via microrheology: tracking the motion of a colloidal probe particle in order to measure the viscoelastic response of the embedding material. The passive probe particle with size R is pulled through the suspension by an external force Fext, which causes it to translate at some speed Uprobe. The bath is comprised of a Newtonian solvent with viscosity ηs and a dilute dispersion of active Brownian particles (ABPs) with size a, characteristic swim speed U0, and a reorientation time τR. The motion of the probe distorts the suspension microstructure, so the bath exerts a reactive force on the probe. In a passive suspension, the degree of distortion is governed by the Péclet number, Pe = Fext/(kBT/a), the ratio of the external force to the thermodynamic restoring force of the suspension. In active suspensions, however, the relevant parameter is Ladv/l = UprobeτR/U0τR∼Fext/Fswim, where Fswim = ζU0 is the swim force that propels the ABPs (ζ is the Stokes drag on a swimmer). When the external forces are weak, Ladv≪l, the autonomous motion of the bath particles leads to "swim-thinning," though the effective suspension viscosity is always greater than ηs. When advection dominates, Ladv≫l, we recover the familiar behavior of the microrheology of passive suspensions. The non-Newtonian behavior for intermediate values of Ladv/l is determined by l/Rc = U0τR/Rc-the ratio of the swimmer's run length l to the geometric length scale associated with interparticle interactions Rc = R + a. The results in this manuscript are approximate as they are based on numerical solutions to mean-field equations that describe the motion of the active bath particles.

Diffusion and flow in complex liquids.

Thermal motion of particles and molecules in liquids underlies many chemical and biological processes. Liquids, especially in biology, are complex due to structure at multiple relevant length scales. While diffusion in homogeneous simple liquids is well understood through the Stokes-Einstein relation, this equation fails completely in describing diffusion in complex media. Modeling, understanding, engineering and controlling processes at the nanoscale, most importantly inside living cells, requires a theoretical framework for the description of viscous response to allow predictions of diffusion rates in complex fluids. Here we use a general framework with the viscosity η(k) described by a function of wave vector in reciprocal space. We introduce a formulation that allows one to relate the rotational and translational diffusion coefficients and determine the viscosity η(k) directly from experiments. We apply our theory to provide a database for rotational diffusion coefficients of proteins/protein complexes in the bacterium E. coli. We also provide a database for the diffusion coefficient of proteins sliding along major grooves of DNA in E. coli. These parameters allow predictions of rate constants for association of proteins. In addition to constituting a theoretical framework for description of diffusion of probes and viscosity in complex fluids, the formulation that we propose should decrease substantially the cost of numerical simulations of transport in complex media by replacing the simulation of individual crowding particles with a continuous medium characterized by a wave-length dependent viscosity η(k).

Alternative Frictional Model for Discontinuous Shear Thickening of Dense Suspensions: Hydrodynamics.

A consensus has emerged that a constraint to rotational or sliding motion of particles in dense suspensions under flow is the genesis of the discontinuous shear thickening (DST) phenomenon. We show that tangential fluid lubrication interactions due to finite-sized asperities on particle surfaces effectively provide these constraints, changing the dynamics of particle motion. By explicitly resolving for the surface roughness of particles, we show that, while smooth particles exhibit continuous shear thickening, purely hydrodynamic interactions in rough particles result in DST. In contrast to the frictional contact model, the hydrodynamic model predicts negative first and second normal stress differences for dense suspensions in the shear thickened state.

Fluctuation-dissipation in active matter.

In a colloidal suspension at equilibrium, the diffusive motion of a tracer particle due to random thermal fluctuations from the solvent is related to the particle's response to an applied external force, provided this force is weak compared to the thermal restoring forces in the solvent. This is known as the fluctuation-dissipation theorem (FDT) and is expressed via the Stokes-Einstein-Sutherland (SES) relation D = kBT/ζ, where D is the particle's self-diffusivity (fluctuation), ζ is the drag on the particle (dissipation), and kBT is the thermal Boltzmann energy. Active suspensions are widely studied precisely because they are far from equilibrium-they can generate significant nonthermal internal stresses, which can break the detailed balance and time-reversal symmetry-and thus cannot be assumed to obey the FDT a priori. We derive a general relationship between diffusivity and mobility in generic colloidal suspensions (not restricted to near equilibrium) using generalized Taylor dispersion theory and derive specific conditions on particle motion required for the FDT to hold. Even in the simplest system of active Brownian particles (ABPs), these conditions may not be satisfied. Nevertheless, it is still possible to quantify deviations from the FDT and express them in terms of an effective SES relation that accounts for the ABPs conversion of chemical into kinetic energy.

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 â†’ ∞ this is the familiar swim pressure [Takatori et al., Phys. Rev. Lett., 2014, 113, 1-5].

Microstructures and mechanics in the colloidal film drying process.

We use Brownian Dynamics (BD) simulations and continuum models to study the microstructures and mechanics in the colloidal film drying process. Colloidal suspensions are compressed between a planar moving interface and a stationary substrate. In the BD simulations, we develop a new Energy Minimization Potential-Free (EMPF) algorithm to enforce the hard-sphere potential in confined systems and to accurately measure the stress profile. The interface moves either at a constant velocity Uw or via a constant imposed normal stress Σe. Comparing the interface motions to the particle Brownian motion defines the Péclet numbers PeU = Uwa/d0 and PeΣ = Σea3/kBT, respectively, where d0 = kBT/ζ with kBT the thermal energy scale, ζ the single-particle resistance, and a the particle radius. With a constant interface velocity, thermodynamics drives the suspension behavior when PeU ≪ 1, and homogeneous crystallization appears when the gap spacing between the two boundaries pushes the volume fraction above the equilibrium phase boundary. In contrast, when PeU ≫ 1, local epitaxial crystal growth appears adjacent to the moving interface even for large gap sizes. Interestingly, the most amorphous film microstructures are found at moderate PeU. The film stress profile develops sharp transitions and becomes step-like with growing Péclet number. With a constant imposed stress, the interface stops moving as the suspension pressure increases and the microstructural and mechanical behaviors are similar to the constant velocity case. Comparison with the simulations shows that the model accurately captures the stress on the moving interface, and quantitatively resolves the local stress and volume fraction distributions for low to moderate Péclet numbers. This work demonstrates the critical role of interface motion on the film microstructures and stresses.

The behavior of active diffusiophoretic suspensions: An accelerated Laplacian dynamics study.

Diffusiophoresis is the process by which a colloidal particle moves in response to the concentration gradient of a chemical solute. Chemically active particles generate solute concentration gradients via surface chemical reactions which can result in their own motion - the self-diffusiophoresis of Janus particles - and in the motion of other nearby particles - normal down-gradient diffusiophoresis. The long-range nature of the concentration disturbance created by a reactive particle results in strong interactions among particles and can lead to the formation of clusters and even coexisting dense and dilute regions often seen in active matter systems. In this work, we present a general method to determine the many-particle solute concentration field allowing the dynamic simulation of the motion of thousands of reactive particles. With the simulation method, we first clarify and demonstrate the notion of "chemical screening," whereby the long-ranged interactions become exponentially screened, which is essential for otherwise diffusiophoretic suspensions would be unconditionally unstable. Simulations show that uniformly reactive particles, which do not self-propel, form loosely packed clusters but no coexistence is observed. The simulations also reveal that there is a stability threshold - when the "chemical fuel" concentration is low enough, thermal Brownian motion is able to overcome diffusiophoretic attraction. Janus particles that self-propel show coexistence, but, interestingly, the stability threshold for clustering is not affected by the self-motion.

Constant Stress and Pressure Rheology of Colloidal Suspensions.

We study the constant stress and pressure rheology of dense hard-sphere colloidal suspensions using Brownian dynamics simulation. Expressing the flow behavior in terms of the friction coefficient-the ratio of shear to normal stress-reveals a shear arrest point from the collapse of the rheological data in the non-Brownian limit. The flow curves agree quantitatively (when scaled) with the experiments of Boyer et al. [Phys. Rev. Lett. 107, 188301 (2011)]. Near suspension arrest, both the shear and the incremental normal viscosities display a universal power law divergence. This work shows the important role of jamming on the arrest of colloidal suspensions and illustrates the care needed when conducting and analyzing experiments and simulations near the flow-arrest transition.

Classical Liquids in Fractal Dimension.

We introduce fractal liquids by generalizing classical liquids of integer dimensions d=1,2,3 to a noninteger dimension dl. The particles composing the liquid are fractal objects and their configuration space is also fractal, with the same dimension. Realizations of our generic model system include microphase separated binary liquids in porous media, and highly branched liquid droplets confined to a fractal polymer backbone in a gel. Here, we study the thermodynamics and pair correlations of fractal liquids by computer simulation and semianalytical statistical mechanics. Our results are based on a model where fractal hard spheres move on a near-critical percolating lattice cluster. The predictions of the fractal Percus-Yevick liquid integral equation compare well with our simulation results.

The swim force as a body force.

Net (as opposed to random) motion of active matter results from an average swim (or propulsive) force. It is shown that the average swim force acts like a body force - an internal body force. As a result, the particle-pressure exerted on a container wall is the sum of the swim pressure [Takatori et al., Phys. Rev. Lett., 2014, 113, 028103] and the 'weight' of the active particles. A continuum description is possible when variations occur on scales larger than the run length of the active particles and gives a Boltzmann-like distribution from a balance of the swim force and the swim pressure. Active particles may also display 'action at a distance' and accumulate adjacent to (or be depleted from) a boundary without any external forces. In the momentum balance for the suspension - the mixture of active particles plus fluid - only external body forces appear.

A theory for the phase behavior of mixtures of active particles.

Systems at equilibrium like molecular or colloidal suspensions have a well-defined thermal energy kBT that quantifies the particles' kinetic energy and gauges how "hot" or "cold" the system is. For systems far from equilibrium, such as active matter, it is unclear whether the concept of a "temperature" exists and whether self-propelled entities are capable of thermally equilibrating like passive Brownian suspensions. Here we develop a simple mechanical theory to study the phase behavior and "temperature" of a mixture of self-propelled particles. A mixture of active swimmers and passive Brownian particles is an ideal system for discovery of the temperature of active matter and the quantities that get shared upon particle collisions. We derive an explicit equation of state for the active/passive mixture to compute a phase diagram and to generalize thermodynamic concepts like the chemical potential and free energy for a mixture of nonequilibrium species. We find that different stability criteria predict in general different phase boundaries, facilitating considerations in simulations and experiments about which ensemble of variables are held fixed and varied.