Quorum-sensing active particles with discontinuous motility

We develop a dynamic mean-field theory for polar active particles that interact through a self-generated field, in particular one generated through emitting a chemical signal. While being a form of chemotactic response, it is different from conventional chemotaxis in that particles discontinuously change their motility when the local concentration surpasses a threshold. The resulting coupled equations for density and polarization are linear and can be solved analytically for simple geometries, yielding inhomogeneous density profiles. Specifically, here we consider a planar and circular interface. Our theory thus explains the observed coexistence of dense aggregates with an active gas. There are, however, differences from the more conventional picture of liquid-gas coexistence based on a free energy, most notably the absence of a critical point. We corroborate our analytical predictions by numerical simulations of active particles under confinement and interacting through volume exclusion. Excellent quantitative agreement is reached through an effective translational diffusion coefficient. We finally show that an additional response to the chemical gradient direction is sufficient to induce vortex clusters. Our results pave the way to engineer motility responses in order to achieve aggregation and collective behavior even at unfavorable conditions.

Figure 1

Discontinuous particle response to a self-produced chemical concentration field $c(\mathbf{r},t)$ (yellow). (a) Motility response $v\left(c\right)$: Particles propel with velocity $v_0$ when they sense concentrations smaller than the threshold $\overline{c}$ and are passive ($v=0$) when their measured concentration surpasses $\overline{c}$. This leads to the formation of an active and a passive domain with the boundary set by $c(\mathbf{r},t)=\overline{c}$. (b) Motility and orientational response $\chi \left(c\right)$: Particles switch from high speed $v^{>}$ to small speed $v^{<}$ and orient perpendicular to $\mathbf{\nabla }c$ when exceeding $\overline{c}$.

Figure 2

Continuity condition. (a) Domain $\mathcal{C}$ inside which $c>\overline{c}$ is bounded by the curve $\partial \mathcal{C}$ defined through $c=\overline{c}$. Along this curve, the speed $v\left(c\right)$ and the coupling $\chi \left(c\right)$ are discontinuous. (b) Zoom to part of the boundary curve $\partial \mathcal{C}$ with integration area $\delta A$.

Figure 3

Planar interface. (a) Passive (${\rho }_c$) and active gas (${\rho }_g$) densities as a function of propulsion speed for $\overline{c}=2c_0$ and $\lambda =0.45\ell$ with respect to a uniform system at density ${\rho }_0$. There is a minimal speed $v_{\text{min}}$ below which the system remains a homogeneous active gas. The inset shows the profiles $\rho \left(x\right)$ and $p\left(x\right)$ at $v_0=v_{*}$. (b) Corresponding $v_0–\overline{c}$ phase diagram. As the threshold $\overline{c}$ is increased, $v_{\text{min}}$ rises (upper green line). For thresholds below the concentration of the uniform system $c_0$, another homogeneous region exists in which all particles become passive.

Figure 4

Paradigms for inhomogeneous systems. (a) Conventional phase diagram for liquid-gas phase separation as observed for active Brownian particles. The two-phase region is bounded by the binodal (thick line), which terminates in a critical point (CP). Arrows show two global densities yielding the same coexisting densities (but different relative size of dense to dilute regions). (b) Phase diagram for discontinuous motility response at fixed threshold $\overline{c}$. Notable is the absence of a critical point. For ${\rho }_0>\overline{\rho }$, the system becomes homogeneous and passive. The dashed line is the prediction Eq. (25) for the maximal cluster density.

Figure 5

Circular clusters. (a) Comparison of radial density profile $\rho \left(r\right)$ from simulations (blue) and theory [Eq. (28)]: unmodified (gray line) and with effective diffusion coefficient $D_{\text{eff}}=10.2D_0$ (black line) for $v_0=20$, $\lambda =10$, $\overline{c}=1.1c_0$. The inset shows a corresponding simulation snapshot with a passive cluster (blue) surrounded by an active gas (red). (b) Cluster and gas densities versus threshold $\overline{c}$ at $\lambda =10$ for different propulsion speeds $v_0$ (symbols: simulations, lines: theory with fitted $D_{\text{eff}}$). At higher $\overline{c}$ we do not observe the formation of stable clusters. Clusters also exist at lower $\overline{c}$ than shown here, but in that case the gas density is not measurable reliably because of the influence of the boundary. (c) Cluster size $r_c$ versus $\overline{c}$ for different $v_0$ [same parameters as in (b)]. Standard errors are smaller than symbol size.

Figure 6

Interfacial width and effective diffusion. (a) Interfacial width $\omega$ between cluster and gas versus propulsion speed $v_0$ for different decay lengths $\lambda$. Error bars show the standard errors. (b) Fitted effective diffusion coefficients $D_{\text{eff}}$ for different $\lambda$ [same colors as in (a)] as double-logarithmic plot. The data collapse onto a single curve when rescaled by $\lambda$ with exponent $\alpha \simeq 0.13$. For narrow interfaces, the effective diffusion coefficient becomes independent of $\lambda$ and rises as $D_{\text{eff}}\sim {(\omega /v_0)}^{2/5}$ (left solid line), whereas for broad interfaces it scales $\sim {(\omega /v_0)}^{-1/3}$ (right solid line). Error bars show standard deviations of fits.

Figure 7

Cluster stability under confinement (at $\lambda =10$). (a) Concentration at cluster boundary $c\left(r_c\right)$ [solution of Eq. (34)] and (b) coexisting densities as function of cluster size $r_c$ for $v_0=20$. For a given threshold $\overline{c}>c_0$ (dotted line $\overline{c}=1.1c_0$) there are two solutions for $r_c$ with different coexisting densities ${\rho }_{c,g}$ (in the simulations the solution with larger $r_c$ is selected). The maximum ${\overline{c}}_{\text{max}}$ sets the maximal threshold at which passive clusters can exist. For smaller speeds (shown is $v_0=5$) ${\overline{c}}_{\text{max}}$ decreases. The value of $c(r_c=R)$ sets the minimum threshold ${\overline{c}}_{\text{min}}$ below which the system is entirely passive. (c) Cluster and gas densities for different $\overline{c}$ as a function $v_0$ from simulations (symbols) and theory (lines). In the simulations, there exists a minimal speed $v_{\text{min}}$ below which no stable clusters exist. The theory predictions for $v_{\text{min}}$ [determined by ${\overline{c}}_{\text{max}}\left(v_0\right)=\overline{c}$] are shown as dotted lines. For $\overline{c}=1.7,2.0c_0$ the system was initialized in a crystalline configuration. (d) Phase diagram $v_0–\overline{c}$ from simulations. Blue: Passive homogeneous system (defined by ${\rho }_c<1.02{\rho }_0$). Green: Passive clusters surrounded by an active gas form from a random starting configuration. Orange: Metastable clusters, which only form when particles are initialized as a crystal. Red: No clusters form; the system is a homogeneous active gas. Upper black line: Maximal threshold ${\overline{c}}_{\text{max}}\left(v_0\right)$ from theory. Above the dashed line, the theory predicts clusters denser than dense packing. Lower black line: Minimal threshold ${\overline{c}}_{\text{min}}$ from Eq. (36).

Figure 8

Vortex clusters [$\lambda =20$, $\overline{c}=0.86c_0$, $v^{>}=30$, $v^{<}=\frac{1}{8}{v}^{>}$]. (a) Snapshot of the system with coupling strength $\chi =-32.7D_r$. Particles are colored by their angular alignment, $e_{\theta }=\mathbf{e}·{\mathbf{e}}_{\theta }$. (b) Schematic illustration of the alignment mechanism. On average the concentration gradient $\mathbf{\nabla }c$ (red) points toward the particles' center of mass ${\mathbf{r}}_{\text{c.m.}}$. The torque of strength $\chi$ (brown) aligns the particle orientation $\mathbf{e}$ perpendicular to $\mathbf{\nabla }c$ (blue), inducing circular particle motion around ${\mathbf{r}}_{\text{c.m.}}$ (dashed line). The direction of $\mathbf{\nabla }c$ fluctuates around $-{\mathbf{e}}_r$ (red cone, angle $\delta$), causing the preferred orientation to fluctuate as well (blue cone). (c) Radial profile of concentration $c$ and magnitude of radial component of its gradient $|{\partial }_{r}c|$ inside the cluster for $\chi =-32.7D_r$. Inset: Distribution of angle $\delta$ for different distances $r$ from center of mass (indicated by Roman numerals). (d) Radial profiles of density $\rho$, (e) average angular orientation $\langle e_{\theta }\rangle$, and (f) radial orientation $\langle e_r\rangle$ for different $\chi$ [labels in (e) refer to (c)].

Figure 9

Maximum $|\langle e_{\theta }\rangle {|}_{\text{max}}$ of the average angular particle orientation $|\langle e_{\theta }\rangle |\left(r\right)$ versus torque strength $\left|\chi \right|$ for different velocities $v^{<}$ in the cluster. Parameters are $\lambda =20$, $v^{>}=30$, and $\overline{c}=0.86c_0$ (standard errors are smaller than symbol size). The $v^{<}$-independent theory prediction based on Eq. (47) is shown in black. For higher $\left|\chi \right|$, $|\langle e_{\theta }\rangle {|}_{\text{max}}$ saturates and the theory line decays to zero (data not shown).