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.

The influence of molecular design on structure-property relationships of a supramolecular polymer prodrug.

Supramolecular self-assemblies of hydrophilic macromolecules functionalized with hydrophobic, structure-directing components have long been used for drug delivery. In these systems, loading of poorly soluble compounds is typically achieved through physical encapsulation during or after formation of the supramolecular assembly, resulting in low encapsulation efficiencies and limited control over release kinetics, which are predominately governed by diffusion and carrier degradation. To overcome these limitations, amphiphilic prodrugs that leverage a hydrophobic drug as both the therapeutic and structure-directing component can be used to create supramolecular materials with higher loading and controlled-release kinetics using biodegradable or enzymatically cleavable linkers. Here, we report the design, synthesis, and characterization of a library of supramolecular polymer prodrugs based on poly(ethylene glycol) (PEG) and the proregenerative drug 1,4-dihydrophenonthrolin-4-one-3-carboxylic acid (DPCA). Structure-property relationships were elucidated through experimental characterization of prodrug behavior in both the wet and dry states using scattering techniques and electron microscopy and corroborated by coarse-grained modeling. Molecular architecture and the hydrophobic-to-hydrophilic ratio of PEG-DPCA conjugates strongly influenced their physical state in water, ranging from fully soluble to supramolecular spherical assemblies and nanofibers. Molecular design and supramolecular structure, in turn, were shown to dramatically alter hydrolytic and enzymatic release and cellular transport of DPCA. In addition to potentially expanding therapeutic options for DPCA through control of supramolecular assemblies, the design principles elaborated here may inform the development of other supramolecular prodrugs based on hydrophobic small-molecule compounds.

Remembering the Work of Phillip L. Geissler: A Coda to His Scientific Trajectory.

Phillip L. Geissler made important contributions to the statistical mechanics of biological polymers, heterogeneous materials, and chemical dynamics in aqueous environments. He devised analytical and computational methods that revealed the underlying organization of complex systems at the frontiers of biology, chemistry, and materials science. In this retrospective we celebrate his work at these frontiers.

Oversaturating Liquid Interfaces with Nanoparticle-Surfactants.

Assemblies of nanoparticles at liquid interfaces hold promise as dynamic "active" systems when there are convenient methods to drive the system out of equilibrium via crowding. To this end, we show that oversaturated assemblies of charged nanoparticles can be realized and held in that state with an external electric field. Upon removal of the field, strong interparticle repulsive forces cause a high in-plane electrostatic pressure that is released in an explosive emulsification. We quantify the packing of the assembly as it is driven into the oversaturated state under an applied electric field. Physiochemical conditions substantially affect the intensity of the induced explosive emulsification, underscoring the crucial role of interparticle electrostatic repulsion.

Boundary design regulates the diffusion of active matter in heterogeneous environments.

Physical boundaries play a key role in governing the overall transport properties of nearby self-propelled particles. In this work, we develop dispersion theories and conduct Brownian dynamics simulations to predict the coupling between surface accumulation and effective diffusivity of active particles in boundary-rich media. We focus on three models that are well-understood for passive systems: particle transport in (i) an array of fixed volume-excluding obstacles; (ii) a pore with spatially heterogeneous width; and (iii) a tortuous path with kinks and corners. While the impact of these entropic barriers on passive particle transport is well established, we find that these classical models of porous media flows break down due to the unique interplay between activity and the microstructure of the internal geometry. We study the activity-induced slowdown of effective diffusivity by formulating a Smoluchowski description of long-time self diffusivity which contains contributions from the density and fluctuation fields of the active particles. Particle-based and finite element simulations corroborate this perspective and reveal important nonequilibrium considerations of active transport.

Phase coexistence implications of violating Newton's third law.

Newton's third law, action = reaction, is a foundational statement of classical mechanics. However, in natural and living systems, this law appears to be routinely violated for constituents interacting in a nonequilibrium environment. Here, we use computer simulations to explore the macroscopic phase behavior implications of breaking microscopic interaction reciprocity for a simple model system. We consider a binary mixture of attractive particles and introduce a parameter that is a continuous measure of the degree to which interaction reciprocity is broken. In the reciprocal limit, the species are indistinguishable, and the system phase separates into domains with distinct densities and identical compositions. Increasing nonreciprocity is found to drive the system to explore a rich assortment of phases, including phases with strong composition asymmetries and three-phase coexistence. Many of the states induced by these forces, including traveling crystals and liquids, have no equilibrium analogs. By mapping the complete phase diagram for this model system and characterizing these unique phases, our findings offer a concrete path forward toward understanding how nonreciprocity shapes the structures found in living systems and how this might be leveraged in the design of synthetic materials.

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.

Phase Diagram of Active Brownian Spheres: Crystallization and the Metastability of Motility-Induced Phase Separation.

Motility-induced phase separation (MIPS), the phenomenon in which purely repulsive active particles undergo a liquid-gas phase separation, is among the simplest and most widely studied examples of a nonequilibrium phase transition. Here, we show that states of MIPS coexistence are in fact only metastable for three-dimensional active Brownian particles over a very broad range of conditions, decaying at long times through an ordering transition we call active crystallization. At an activity just above the MIPS critical point, the liquid-gas binodal is superseded by the crystal-fluid coexistence curve, with solid, liquid, and gas all coexisting at the triple point where the two curves intersect. Nucleating an active crystal from a disordered fluid, however, requires a rare fluctuation that exhibits the nearly close-packed density of the solid phase. The corresponding barrier to crystallization is surmountable on a feasible timescale only at high activity, and only at fluid densities near maximal packing. The glassiness expected for such dense liquids at equilibrium is strongly mitigated by active forces, so that the lifetime of liquid-gas coexistence declines steadily with increasing activity, manifesting in simulations as a facile spontaneous crystallization at extremely high activity.

Mechanisms of Diffusion in Associative Polymer Networks: Evidence for Chain Hopping.

Networks assembled by reversible association of telechelic polymers constitute a common class of soft materials. Various mechanisms of chain migration in associative networks have been proposed; yet there remains little quantitative experimental data to discriminate among them. Proposed mechanisms for chain migration include multichain aggregate diffusion as well as single-chain mechanisms such as "walking" and "hopping", wherein diffusion is achieved by either partial ("walking") or complete ("hopping") disengagement of the associated chain segments. Here, we provide evidence that hopping can dominate the effective diffusion of chains in associative networks due to a strong entropic penalty for bridge formation imposed by local network structure; chains become conformationally restricted upon association with two or more spatially separated binding sites. This restriction decreases the effective binding strength of chains with multiple associative domains, thereby increasing the probability that a chain will hop. For telechelic chains this manifests as binding asymmetry, wherein the first association is effectively stronger than the second. We derive a simple thermodynamic model that predicts the fraction of chains that are free to hop as a function of tunable molecular and network properties. A large set of self-diffusivity measurements on a series of model associative polymers finds good agreement with this model.

Analysis and Control of Chain Mobility in Protein Hydrogels.

Coiled-coil domains can direct the assembly of protein block copolymers into physically cross-linked, viscoelastic hydrogels. Here, we describe the use of fluorescence recovery after photobleaching (FRAP) to probe chain mobility in reversible hydrogels assembled from engineered proteins bearing terminal coiled-coil domains. We show that chain mobility can be related to the underlying dynamics of the coiled-coil domains by application of a three-state "hopping" model of chain migration. We further show that genetic programming allows the effective mobility of network chains to be varied 500-fold through modest changes in protein sequence. Destabilization of the coiled-coil domains by site-directed mutagenesis increases the effective diffusivity of probe chains. Conversely, probe mobility is reduced by expanding the hydrophobic surface area of the coiled-coil domains through introduction of the bulky leucine surrogate homoisoleucine. Predictions from the three-state model imply asymmetric sequential binding of the terminal domains. Brownian Dynamics simulations suggest that binding asymmetry is a general feature of reversible gels, arising from a loss in entropy as chains transition to a conformationally restricted bridged state.

Shear-Induced Heterogeneity in Associating Polymer Gels: Role of Network Structure and Dilatancy.

We study associating polymer gels under steady shear using Brownian dynamics simulation to explore the interplay between the network structure, dynamics, and rheology. For a wide range of flow rates, we observe the formation of shear bands with a pronounced difference in shear rate, concentration, and structure. A striking increase in the polymer pressure in the gradient direction with shear, along with the inherently large compressibility of the gels, is shown to be a crucial factor in destabilizing homogeneous flow through shear-gradient concentration coupling. We find that shear has only a modest influence on the degree of association, but induces marked spatial heterogeneity in the network connectivity. We attribute the increase in the polymer pressure (and polymer mobility) to this structural reorganization.

Self-Propulsion by Directed Explosive Emulsification.

An active droplet system, programmed to repeatedly move autonomously at a specific velocity in a well-defined direction, is demonstrated. Coulombic energy is stored in oversaturated interfacial assemblies of charged nanoparticle-surfactants by an applied DC electric field and can be released on demand. Spontaneous emulsification is suppressed by an increase in the stiffness of the oversaturated assemblies. Rapidly removing the field releases the stored energy in an explosive event that propels the droplet, where thousands of charged microdroplets are ballistically ejected from the surface of the parent droplet. The ejection is made directional by a symmetry breaking of the interfacial assembly, and the combined interaction force of the microdroplet plume on one side of the droplet propels the droplet distances tens of times its size, making the droplet active. The propulsion is autonomous, repeatable, and agnostic to the chemical composition of the nanoparticles. The symmetry-breaking in the nanoparticle assembly controls the microdroplet velocity and direction of propulsion. This mechanism of droplet propulsion will advance soft micro-robotics, establishes a new type of active matter, and introduces new vehicles for compartmentalized delivery.

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.

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.

Swimming to Stability: Structural and Dynamical Control via Active Doping.

External fields can decidedly alter the free energy landscape of soft materials and can be exploited as a powerful tool for the assembly of targeted nanostructures and colloidal materials. Here, we use computer simulations to demonstrate that nonequilibrium internal fields or forces-forces that are generated by driven components within a system-in the form of active particles can precisely modulate the dynamical free energy landscape of a model soft material, a colloidal gel. Embedding a small fraction of active particles within a gel can provide a unique pathway for the dynamically frustrated network to circumvent the kinetic barriers associated with reaching a lower free energy state through thermal fluctuations alone. Moreover, by carefully tuning the active particle properties (the propulsive swim force and persistence length) in comparison to those of the gel, the active particles may induce depletion-like forces between the constituent particles of the gel despite there being no geometric size asymmetry between the particles. These resulting forces can rapidly push the system toward disparate regions of phase space. Intriguingly, the state of the material can be altered by tuning macroscopic transport properties such as the solvent viscosity. Our findings highlight the potential wide-ranging structural and kinetic control facilitated by varying the dynamical properties of a remarkably small fraction of driven particles embedded in a host material.

Aggregation behavior of rod-coil-rod triblock copolymers in a coil-selective solvent.

Recent experiments have reported that the self-assembly of conjugated polymers mimicking rod-coil-rod triblock copolymers (BCPs) in selective solvents exhibits unique aggregate morphologies. However, the nature of the arrangement of the polymers within the aggregates and the spatial organization of the aggregates remain an unresolved issue. We report the results of coarse-grained Langevin dynamics simulations, which investigated the self-assembly behavior of rod-coil-rod BCPs in a coil-selective solvent. We observe a rapid formation of cylindrically shaped multichain clusters. The initial stages of formation of the aggregates was seen to be independent of the strength of the solvent selectivity. However, for higher solvent selectivities, the clusters were seen to merge into larger units at later stages. A reduction in rod to coil block ratio was observed to decrease the size and number of clusters. In the limit of a highly concentrated solution, we observe the formation of a networked system of distinct clusters, which however retain the cylindrical arrangement observed at lower polymer concentrations.