Pressure continues to be a topic of fundamental interest in active matter. An interest motivated, in part, by the tantalizing prospect that mechanical equations-of-state (EoS) will play as important of a role in active matter as they do in equilibrium theory. In fact, pressure has already proven to be a salient metric for rationalizing the often-complex behavior of active suspensions ranging from instabilities exhibited by expanding bacterial droplets, vesicles filled with active particles, active depletion, and the dynamics of colloidal gels, membranes, and polymers immersed in a bath of active colloids. The burgeoning of research on the nature of pressure in active systems has not only contributed to progress in understanding the phenomenology of active systems, but has also played a central role in assessing the validity of new theoretical concepts for nonequilibrium systems. There is no better example than the intense focus on the development of nonequilibrium theories for the phase behavior of active particles – a crucial ingredient for many is an EoS.
Pressure in active systems remains difficult to characterize analytically in all but the simplest cases. These challenges can be attributed to a unique nonlocal contribution to the pressure referred to as the swim pressure. This nontraditional source of stress is entirely rooted in trajectory space and unraveling its concentration dependence necessitates the difficult task of understanding how particle trajectories are altered by many-body correlations. The driving force for much of the novel behaviors observed in active systems, including the phenomena of motility-induced phase separation (MIPS), is due to this inherently nonequilibrium contribution to the pressure.
The swim pressure emerges from the persistent and time-irreversible motion generated by the non-conservative self-propelling force of active particles. Within the context of the active Brownian particle (ABP) model a popular minimal model for studying active colloids particles self-propel at a constant speed U0 and undergo Brownian rotational diffusion with a characteristic reorientation time τR. At long times, an ideal ABP executes a random walk with a trajectory correlation length l0 = U0 τR , which we call the intrinsic run-length. A dilute system of ABPs with number density n exerts a swim pressure directly proportional to this in-trinsic run-length, equation. For more concentrated systems, interparticle interactions reduce the tra jectory correlation length, resulting in a density-dependent effective run-length l < l0 and a swim pressure of Πs ∼ nlU0. In the limit of maximal-packing, ABPs are prevented from executing their athermal random walk resulting in a vanishingly small effective run-length (and swim pressure) that should be independent of l0 and depend only on geometric packing constraints. This physical expectation for the swim pressure is currently not reflected in the literature.
This simple physical picture [see Fig. 1] suggests the existence of a concentration scale that controls the crossover between the dynamically distinct dilute and jammed regimes of active suspensions. In this Letter, using large-scale computer simulations, we identify this new run-length dependent concentration φ∗, which exhibits a number of compelling analogies to the powerful concept of the overlap concentration of equilibrium polymer solutions. Using φ∗ and concentration scales associated with colloidal jamming, the mechanical equation-of-state for ABPs can be collapsed onto a single curve which contains a number of previously overlooked features, the inclusion of which qualitatively alters previous predictions of the phase behavior of ABPs. Moreover, the use of a dynamic overlap concentration scale may prove to be of great utility in the construction of accurate nonequilibrium equations-of-state.
Simulation Model and the EoS.– In an effort to unpack the concentration dependence of the pressure, we focus on the simplest and most widely studied active system: purely-repulsive active Brownian disks. By introducing a small degree of polydispersity in disk size, we ensure the system remains disordered for all concentrations. This is in contrast to monodisperse active disks, which exhibit a order-disorder phase transition with a complex dependence on activity. The small degree of polydispersity allows us to isolate the disordered branch of the equations-of-state, which not only will serve as a useful reference system for active fluids with more complicated interactions, but allows for the quantitative study of the interplay between activity and concentration.
We consider a two dimensional system of overdamped disks at fixed number density n where each particle experiences a drag force ζ U , a conservative interparticle force F c and an active self-propelling force F a = ζU0q. The orientation of each particle q is independent and obeys diffusive rotary dynamics with a characteristic reorientation time τR. The sum of the forces results in a simple equation-of-motion for the particle velocity: equation. The interparticle force F c arises from a Weeks-Chandler-Anderson (WCA) potential characterized by a potential depth ε and an average Lennard-Jones diameter d. Particle diameters are drawn from a normal distribution with a standard deviation of 0.1d. Importantly, as our active force is of finite amplitude, a sufficiently strong choice for the repulsive force F c will mimic a true hard-particle potential. A choice of ε/(ζU0d) = 100 is found to result in hard disk statistics with an effective average particle diameter of 21/6d. In this hard-disk limit, the state of our system is independent of the amplitude of the active force and is fully described by two geometric parameters: the area fraction φ = nπ(21/6d)2/4 and the dimensionless intrinsic run-length l0/d. Using the GPU-enabled HOOMD-blue software package, all simulations were conducted with 40,000 particles and run for a minimum duration of 5,000 d/U0.
The total mechanical pressure Π has two contributions: the collisional pressure Πc arising from conservative interparticle interactions and the swim pressure Πs generated by the active force. The collisional pressure follows from the standard micromechanical virial for conservative interactions Πc = n⟨x · F c ⟩/2 where x is the particle position and ⟨...⟩ denotes an average over all particles. At steady-state, Πs can be written in the impulse form: equation. For an isotropic system of active particles free of aligning interaction, Eq. (2) is the pressure that the suspension would exert on a flat torque-free macroscopic boundary. Under these conditions, the total pressure exerted on the surroundings is simply the sum of the collisional and swim pressure Π = Πc + Πs.
Equation (2) allows us to directly probe via simulation the effective run-length of the particles, l ≡ τR ⟨q · U ⟩, which is the true correlation length of a particle's trajectory and a measure of the correlation between the orientation of a particle and the forces (both active and interparticle) acting upon it. Analytical expressions for the effective run-length l require solving a many-body dynamics problem in both position and orientation space, and have been derived in only a few asymptotic limits. Additionally, the nature of l is further obfuscated as direct measurement is limited to a narrow region of (l0 , φ)-state-space where motility-induced phase separation is absent (i.e., outside of the MIPS phase envelope). Paradoxically, the prediction of both the spinodal and binodal requires knowledge of l in regions where it cannot be directly measured. Extrapolation of l into these mechanically forbidden regions requires a complete understanding of the trends in l as the system approaches the critical l0 for MIPS (l0/d ≈ 13). Figure 2(a) presents the complete φ dependence of l in regions of (l0, φ)-statespace where the system is homogeneous.
Our approach for isolating these trends is motivated by two simple observations. First, l becomes increasingly independent of the intrinsic run-length l0 with increasing concentration. In the dilute limit, l = l0 = U0 τR , while in the limit of maximal-packing l ≈ 0. Second, deviations from the dilute scaling prediction of l occur at lower concentrations as l0 is increased. Microrheology calculations for hard-disks show the first-order correction to the effective run-length is given by l ≈ l0(1 − φ) in the limit of large l0. As shown in Fig. 2(b), this expression is in excellent agreement with our data for persistent ABPs (l0 > 3d), and note the previously mentioned trend in the deviation from this dilute scaling as l0 increases.
Dynamic Overlap Concentration.– The dependence of the swim pressure on a single-particle length scale at low concentrations – the intrinsic run-length l0 – and its independence from that length scale at high concentrations suggests a connection to equilibrium polymer solutions. There exist distinct concentration regimes for polymer solutions, corresponding to isolated (dilute) and overlapping chains (semidilute). The thermodynamic equations-of-state in each regime have distinct dependencies on single-chain properties. For example, the osmotic pressure of a dilute polymer solution is inversely proportional to the degree of polymerization N [see Fig. 1]. In the semidilute regime where chains strongly overlap, the individuality of each chain is lost resulting in an osmotic pressure that is independent of the single-chain property N. The overlap concentration φ∗ delineates the dilute and semidilute regimes and is the concentration at which polymer coils begin to overlap. The density of chains at φ∗ is inversely proportional to the effective volume occupied by a single polymer chain V chain.
For ABPs, we seek the characteristic single-particle volume (area in 2D) VABP that plays the analogous role to the single-chain volume V chain for polymer solutions. In 2D, a natural starting point is to consider the area swept out by an ideal ABP in a single reorientation time τR: V ABP ∼ l0d. For concentrations beyond φ∗ ∼ 1/V ABP particle trajectories overlap and collisions prevent ABPs from executing their intrinsic random walk of length l0 [see Fig. 1]. The single-particle volume given by V ABP ∼ l0d is likely an overestimate of the overlap probability. More realistically, we anticipate a weaker dependence on l0 and introduce the more general definitionVABP ∼lλ0d2−λ where0<λ<1isaconstanttobe determined. The resulting overlap concentration φ∗ can be written as: equation.
The dynamic overlap concentration φ∗ proposed in Eq. (3) should delineate dilute and semidilute concentration regimes for all values of l0. Using this criterion as a guide (see Supplemental Material), we identify a significant dependence of the dynamic overlap concentration on the intrinsic run-length with λ ≈ 0.353 for our polydisperse system of active Brownian disks. As shown in Fig. 2(c), rescaling φ with φ∗ allows for a collapse of l in the dilute (φ/φ∗ < 1) and semidilute (φ/φ∗ ≥ 1) regimes. Notably, for each l0 deviations from the collapse begin to occur at a φ∗-independent area fraction φ ≈ 0.72, indicative of an upper-bound for our semidilute regime and the emergence of what we term as the active jamming regime.
As the system transitions to the active jamming regime, the run-length tends to zero as the maximalpacking φm ≈ 0.845 is approached. Furthermore, the run-length becomes increasingly independent of l0 as geometric packing constraints set the scale for particle motion. We find l in the active jamming regime is welldescribed by: equation, where D,E > 0 are constant empirical fitting parameters. As shown by the family of curves in Fig. 2(c), Eq. (4) accurately captures the deviation of l from the semidilute regime as a function of l0 while also capturing our physical asymptotic expectations. With increasing l0, the effective run-length approaches a constant value set by the packing geometry i.e., liml0→∞ l = DE(1 − φ/φm). Previous EoS which expressed l = l0 f (φ) (for large l0 ) fail to capture this physical expectation and, in fact, find the unphysical result of liml0→∞ l = ∞ for all φ unless f(φ) is identically zero.
By identifying φ∗ and the scaling behavior in the active jamming regime, the l0 dependence of the three regions of the swim pressure (dilute, semidilute and active jamming) can be described quantitatively. From this information, it is straight-forward to construct a functional form that captures the full φ and l0 dependencies of the swim pressure. In Fig. 2(d), representative curves for the φ∗-informed swim pressure are provided for a range of l0 above the critical intrinsic run-length for MIPS. For comparison, we include expressions for the swim pressure put forward in previous works. Importantly and in contrast to Refs., the swim pressure derived using φ∗ does not asymptote to a characteristic functional form at large `0. In fact, the dependence on `0 is quite striking but not unexpected given our newly identified concentration regimes.