Nonequilibrium Equation of State in Suspensions of Active Colloids

Active colloids constitute a novel class of materials composed of colloidal-scale particles locally converting chemical energy into motility, mimicking micro-organisms. Evolving far from equilibrium, these systems display structural organizations and dynamical properties distinct from thermalized colloidal assemblies. Harvesting the potential of this new class of systems requires the development of a conceptual framework to describe these intrinsically nonequilibrium systems. We use sedimentation experiments to probe the nonequilibrium equation of state of a bidimensional assembly of active Janus microspheres and conduct computer simulations of a model of self-propelled hard disks. Self-propulsion profoundly affects the equation of state, but these changes can be rationalized using equilibrium concepts. We show that active colloids behave, in the dilute limit, as an ideal gas with an activity-dependent effective temperature. At finite density, increasing the activity is similar to increasing adhesion between equilibrium particles. We quantify this effective adhesion and obtain a unique scaling law relating activity and effective adhesion in both experiments and simulations. Our results provide a new and efficient way to understand the emergence of novel phases of matter in active colloidal suspensions.

Popular Summary

Active matter includes a wide range of systems, from swarms of bacteria to flocks of birds. “Activity” means that each elementary constituent possesses its own components to propel itself by consuming energy. Such active populations, which live far from equilibrium, frequently behave collectively in a manner that deviates significantly from the well-posed rules of equilibrium physics. Accordingly, many efforts in recent years have attempted to create artificial active matter, mimicking their living counterpart, in order to obtain a fundamental understanding of the specific rules controlling the physical behavior of these systems. There are many indications that large assemblies of active particles tend to naturally aggregate to form clusters. Our goal is to quantify this behavior in thermodynamic terms.

We address this question by measuring the “equation of state” of a colony of catalytically active colloidal swimmers, represented by 2-$\mu \mathrm{m}$-diameter Janus microspheres made of gold and platinum. In a dilute regime, those active swimmers behave as an ideal gas, with an activity-dependent effective temperature; the higher the activity, the higher the temperature. By measuring the evolution of active pressure as a function of particle density—from the dilute regime to a high-density solid—we demonstrate the adhesive character of the active swimmers. However, surprisingly, the stickiness of the swimmers increases as the system deviates from equilibrium. One explanation for this finding is the stubborn character of the swimmers, which, upon making contact with another particle, take a long time before returning to the rest of the swarm. We also conduct computer simulations of hard disks to compare the equations of state of the nonequilibrium active particles (the microspheres) and the equilibrium adhesive disks. We find that the degree of clustering is similar for both systems of particles.

Our study paves the way for quantifying nonequilibrium interactions between active particles in order to build a nonequilibrium theoretical framework for active colloids.

Figure 1

Imaging sedimentation profiles. Snapshots of (a) phoretic gold-platinum Janus colloids and (b) self-propelled hard disks under gravity for activity increasing from left to right. (a) Four different activities are shown, corresponding to Péclet numbers $P_e=0$ (passive case), 8, 14, and 20 (from left to right). The inset shows a schematic view of the experimental setup. The scale bar is $20\mu \mathrm{m}$. (b) Simulation snapshots for persistence times $\tau =0$ (passive disks), 1, 10, and 100 (from left to right).

Figure 2

Experimental density profiles. (a) Density profiles for various activities, arbitrarily shifted horizontally so that $z_0$ corresponds to $\varphi =0.2$. Inset: Evolution of $T_{\text{eff}}$ as a function of the Péclet number, together with the theoretical expectation $T_{\text{eff}}/T_0=1+\frac{2}{9}P{e}^2$. (b) Zoom-in on the dilute phase $\varphi \ll 1$ using a semilog scale, with the associated exponential fit, from which we extract the value of the effective temperature.

Figure 3

Equation of state and compressibility factor. Equation of state $P/(k_B{T}_0)$ versus linear density $\varphi$ for (a) experiment and (b) simulations. In (a), the inset zooms in on the dilute phase, where lines represent the best linear fit from which we recover $T_{\text{eff}}$. (c) Compressibility factor versus $\varphi$ for experiment. Black points represent the passive case; the dashed black curve represents the empirical equation of state for hard disks. Dashed colored lines are obtained from Eq. (3). (d) Compressibility factor versus $\varphi$ for simulations with dashed lines obtained from the Percus-Yevick closure for the two-dimensional Baxter model. Experiments: $T_{\mathrm{eff}}/T_0=1$, 5, 15, 34, 47, 62, 87 (bottom to top). Simulations: $\tau =0$, 1, 3, 10, 30, 100, 300, 1000 (bottom to top).

Figure 4

Quantifying motility-induced adhesion. Evolution of adhesive parameter $A$ obtained from the equation of state with the effective temperature obtained in the dilute limit for both experiments (circles) and simulations (squares). The lines correspond to the scaling law $A\sim \sqrt{T_{\text{eff}}/T_0}$, with slightly different prefactors for simulations and experiments.