Synthetic Quorum Sensing and Absorbing Phase Transitions in Colloidal Active Matter

TL;DR

The paper addresses how synthetic quorum sensing, implemented as density-dependent motor switching in Quincke rods, reshapes phase behavior of active matter. The authors combine experiments, a minimal active Brownian particle theory, and simulations to show that quorum sensing generically drives an absorbing phase transition, while steric repulsion can arrest it to yield phase coexistence with flat interfaces and a net interfacial pressure drop. A generalized thermodynamics framework recasts the dynamics as gradient flow of an effective free-energy $\mathcal{G}$ with coexisting densities $\rho_L$ and $\rho_A$ obtained by a common-tangent construction, and predicts a pressure imbalance $P_L-P_A$ across the interface due to interfacial flux $\Delta$. The results are argued to generalize to other adaptive active matter systems, providing a route to design responsive, two-way coupled active materials.

Abstract

Unlike biological active matter that constantly adapt to their environment, the motors of synthetic active particles are typically agnostic to their surroundings and merely operate at constant force. Here, we design colloidal active rods capable of modulating their inner activity in response to crowding, thereby enforcing a primitive form of quorum sensing interactions. Through experiments, simulations, and theory we elucidate the impact of these interactions on the phase behavior of isotropic active matter. We demonstrate that, when conditioned to density, motility regulation can either lead to an absorbing phase transition, where all particles freeze their dynamics, or to atypical phase separation, where flat interfaces supporting a net pressure drop are in mechanical equilibrium. Fully active and fully arrested particles can then form heterogeneous patterns ruled by the competition between quorum sensing and mechanical interactions. Beyond the specifics of motile colloids, we expect our findings to apply broadly to adaptive active matter assembled from living or synthetic units.

Figures (6)

• Figure 1: Motorizing colloidal rods. (a) (Left) Sketch of our microfluidic setup. We first tilt the device to let the rods sediment and accumulate at the tip of the V-shaped channel. We then place the device in the horizontal direction and perform our observations. (Right) Bright Field image of the SU8 rods at density $\rho_0=0.16\,\rm \mu m^{-2}$, and voltage $V=150\,\rm V$. Scale bar: $5\,\rm \mu m$. (b) Quincke instabilities. (i) Electrohydrodynamic instability of a dielectic sphere immersed in a conducting fluid. When applying a DC field, the static configuration is unstable when $E>E_Q$, and the sphere spins around a random axis normal to $\mathbf E$Melcher1969. (ii) When the particle is anisotropic, it points in the direction of $\mathbf E$ when $E$ is small. The rod is static. At large $E$, the particle aligns in a random direction normal to $\mathbf E$ and spins around its long axis. At intermediate fields both states are stable and coexist Cebers2000Brosseau2017 (iii) In the vicinity of a solid surface, rods that are dense or charged enough are driven toward the surface. Their behavior mirrors that of of Quincke spheres. When $E<E_Q$ the rod is static and aligned with the solid surface, above $E_Q$ it rolls along the solid wall in the direction of its short axis (c) Superimposed images of SU-8 colloidal rods (length: $3.7\,\rm \mu m$ diameter: $0.55\,\rm \mu m$) sedimented on a solid electrode for $E/E_{Q}=0,\, 2$. The $\mathbf E$ field points in the direction normal to the images. The color indicates time (Time step $0.1\,\rm s$). The polydispersity in the rod length results in a distribution Quincke thresholds. We define $E_Q$ as the field where the first rod starts rolling. Scale bar: $5\,\rm \mu m$.
• Figure 2: Phase behavior of interacting Quincke rods. (a) Bright field picture of a homogeneous liquid phase ($\rho_0=0.12\,\rm \mu m^{-2}$, $V=150\,\rm V$). The color indicates the local orientation of the rods measured using the method of Rezakhaniha et al. rezakhaniha2012. The active liquid is isotropic. (b) Fluorescence image of the same liquid. Unlike in the gas phase, some rods have switched to a standing state: they stop spinning and point in the vertical direction (bright dots). The shaded circles help locating the arrested rods. We stress that only $25\,\%$ of the rods are fluorescent, see Appendix A. (c) Average fraction of standing rods $\alpha/(\alpha+\beta)$ plotted against the average density $\rho_0$. (d) Fluorescence image of arrested clusters coexisting with an active liquid phase ($\rho_0=0.21\,\rm \mu m^{-2}$, $V=80\,\rm V$). In the clusters, the vast majority of the rods stand in the direction of the $\mathbf E$ field. Note that only 25% of the rods are fluorescent, see Appendix A. (e) Three subsequent pictures showing the capture and release of active rods at the interface between the arrested and active-liquid phases. Motile rods may switch to the standing state when colliding with the clusters. Similarly, arrested rods can resume their self-propulsion and escape the cluster boundary. (f) The area fraction of the arrested phase $\Phi_{A}$ grows when $\rho_0$ increases ($V=150\,\rm V$). Fitting the coexistence points with a straight line allows us to define the two binodals $\rho_{\rm L}$ ($\Phi_A=0)$ and $\rho_{\rm A}$ ($\Phi_A=1)$, corresponding to the vertical shaded lines. (g) At the onset of solidification all the $\Phi_{A}(\rho_0-\rho_{\rm L})$ collapse on a master curve (where the binodal $\rho_L$ is defined in (f), yellow vertical line) (h) Phase diagram of interacting Quincke rods. The three macroscopic regions correspond to the homogeneous active liquid, the homogeneous arrested solid and to the coexistence region.
• Figure 3: Quorum sensing induces phase separation. (a) Density of the active liquid ($\rho_{\rm L}$) and of the arrested phases ($\rho_{\rm A}$) plotted against the average rod density $\rho_0$. Horizontal lines correspond to the binodals determined in Fig. \ref{['Fig2']}f. Voltage: $V=150\,\rm V$. (b) Growth of a a single arrested cluster in a homogeneous active liquid. The typical size $R(t)$ is defined as the square root of the cluster area. $V=109\,\rm V$, $\rho_0 = 0.15\,\rm \mu m^{-2}$. Scale bar: $5\,\rm \mu m$. (c) Growth kinetics showing the classical $R(t)\sim (t-t_0)^{1/3}$ scaling law. (d) Image sequence showing the coarsening patterns upon a quench deep in the coexistence region, from $V=300\,V$ to $V=90\,V$. $\rho_0=0.20\,\rm \mu m^{-2}$. Scale bar: $5\,\rm \mu m$. (e) Hysteresis loops. When the motorization voltage varies periodically the fraction of the arrested phase displays a hysteresis loop. $\rho_0=0.2\,\rm \mu m^{-2}$. The two hysteresis cycles corresponds to different periods $T$ of the applied voltage. The voltage varies from $80\,\rm V$ to $270\,\rm V$. (Left): $T=60\,\rm s$. (Right): $T=300\,\rm s$. (f) Corresponding time series of $\Phi_{A}(t)$ and of $V(t)$. ($T=300\,s)$
• Figure 4: Quorum-sensing-induced absorbing phase transition. (a) Streamplot of the mean-field dynamics of Eq \ref{['eq:MF_rel']}. The dashed gray lines correspond to the dynamics, while solid lines represent the fixed points. For $\rho_0<\bar{\rho}$, the active phase is the only stable attractor (yellow line, $\rho_S=0$). At intermediate densities $\bar{\rho} < \rho_0 < \rho_c$, the fixed points correspond to mixed configurations (teal line). At large densities, $\rho_0>\rho_c$, the static phase is the only stable attractor (black line). (b-d) Snapshots from a simulation of ABPs with density-dependent transition rates (Eqs. \ref{['eq:alpha-rods']} and \ref{['eq:beta-rods']}). To facilitate comparison between experiments and simulations, we plot the rolling rods as golden rectangles and the standing rods as black circles. We show in SM that simulations of rods interacting via actual anisotropic quorum-sensing interactions lead to identical results. When an arrested cluster nucleates, it progressively absorbs the whole system as the particle current ${\bf J} \propto - \nabla_{{\bf r}} \rho_R$. In this panel, the length of the rods corresponds to the interaction radius $\ell_r=5$. Numerical details: see Appendix \ref{['app:numerics']} and SM.
• Figure 5: Competition between quorum sensing and pairwise repulsion in numerical simulations. (a) Coarsening of static domains. (b) Nucleation of a static bubble at $\rho_0=0.24$. (c) Spinodal decomposition at $\rho_0=0.45$. (d) Static area fraction $\Phi_{A}$ as a function of $\rho_0$. The solid vertical line corresponds to the critical density $\rho_c$ beyond which the mean-field dynamics predicts a rapid convergence to a homogeneous static phase. The two dashed vertical lines represent the density of coexisting phases. Yellow diamonds correspond to a fully active gas; teal hexagons to phase coexistence; teal and black squares represent metastable regions where both the arresting phase transition and phase coexistence are observed; black triangles to a static solid. (e) Liquid and solid binodals measured in simulations. We use the same convention for symbols as in panel d. (f) Hysteresis loop for the static area fraction as a function of $\bar{\rho}$, i.e. the threshold beyond which the standing rate $\alpha(\rho)>0$. In simulations, the activity of the system is varied in time via a sawtooth protocol for $\bar{\rho}(t)$ and $v_0(t)$ with period $T$ as shown in the black curve of panel (g). (g) Time evolution of $\Phi_{A}$ under a sawtooth protocol for $\bar{\rho}(t)$ and $v_0(t)$. Numerical details: see Appendix \ref{['app:numerics']} and SM.