Dynamic Instabilities and Pattern Formation in Chemotactic Active Matter

Abstract

Collectives of actively-moving particles can spontaneously segregate into dilute and dense phases through a process known as motility-induced phase separation (MIPS). This captivating phenomenon is well-studied for randomly-moving particles with no directional bias. However, many active systems perform collective chemotaxis -- directed motion along a chemical gradient collectively generated by the particles themselves through consumption or production. Here, we use linear stability analysis, amplitude equations, and numerical simulations to study how MIPS is influenced by collective chemotaxis. We find that chemotaxis can either arrest or entirely suppress MIPS, or give rise to novel dynamic instabilities such as traveling waves and spirals. We predict the stability region of the stationary and oscillatory patterns and identify four types of bifurcation that can arise: pitchfork, saddle-node, infinite period, and supercritical Hopf. We also derive analytical expressions for the amplitude of the pattern and traveling wave velocity, yielding excellent quantitative agreement with simulations. Furthermore, we generalize our model to study particles that either consume or produce chemoattractant or chemorepellent, as well as mixtures of particles with different chemotactic behaviors. By establishing quantitative principles describing the competition between MIPS and chemotaxis, our study helps deepen understanding of the rich physics underlying chemically-responsive active matter systems.

Figures (18)

• Figure 1: Schematic illustration of chemotactic active Brownian particles (ABPs) undergoing motility-induced phase separation. At sufficiently low reorientation Péclet number $\text{Pe}_R$ and moderate particle volume fraction, non-chemotactic ABPs (blue) form dense clusters via MIPS. When particles consume a chemoattractant (yellow), they direct their motion toward higher concentrations and migrate away from depleted clusters. As a result, chemotaxic dispersal competes with MIPS clustering. Producers of chemorepellent exhibit qualitatively similar dispersal.
• Figure 2: Linear stability analysis reveals three distinct regimes of chemotactic MIPS depending on chemical diffusivity and chemotactic strength. (a) Outside the MIPS spinodal region ($\sigma=1$), the uniform state is stable when $\text{Pe}_C'\geq-1$, independent of $\mathrm{Da}'$ and $\alpha'$; stationary instability occurs when $\text{Pe}_C'<-1$. (b) Inside the MIPS spinodal region ($\sigma=-1$), three regimes emerge: stable (dark teal, where $\alpha' \leq \alpha'_\text{crit}$ and $\text{Pe}_C' \geq \text{Pe}'_{C,\text{crit}}$), stationary instability (light green), and oscillatory instability (orange, where $\alpha'>\alpha'_\text{crit}$ and $\text{Pe}_C'>\text{Pe}_{{C,*}}'$). Increasing $\mathrm{Da}'$ (faster chemical uptake) shrinks both instability regions. (c) Representative dispersion relations showing the positive branch of the nondimensionalized eigenvalue $\tilde{\omega}_+$ at $\text{Da}'=0.2$ for the three regimes marked in (b): stable (dot), stationary instability (square), and oscillatory instability (diamond).
• Figure 3: Three-dimensional phase diagram reveals how chemotaxis suppresses MIPS and generates oscillatory instabilities in consumer particles. (a,d) Phase behavior of chemotactic ABPs that consume a chemoattractant in the $\text{Pe}_R-\phi_0-\text{Pe}_C$ phase space at $\text{Da}=0.3$ with $\alpha=1$ (a) and $\alpha=10$ (d). Gray surface marks the stability boundary from linear stability analysis; above it, the homogeneous state is stable, while below it, phase separation occurs. At small $\alpha$ (a), chemotaxis arrests coarsening below the boundary, producing stationary dots and stripes. At large $\alpha$ (d), the colored surface delineates stationary (left) from oscillatory (right) instability; the color represents the value of $\text{Pe}_C$, and the colormap ranges from the minimum to maximum values of $\text{Pe}_C$ of the surface. The red dashed curve marks the transition from one regime to the other ($\text{Pe}_C'=\text{Pe}'_{{C,\text{crit}}}$ and $\alpha' = \alpha'_\text{crit}$). (b-c, e-f) Cross-sections of the $\text{Pe}_R-\phi_0$ phase space at $\text{Pe}_C=0.35$ and $0.8$; black curves are cross-sections through the gray surface, and the green curve in (e) shows a cross-section through the colored surface in (d). We expect a stationary instability below the black curves in (b)-(c) and below the green curve in (e), and an oscillatory instability in the gray shaded region between the black and green curves in (e) and below the black curve in (f). Simulation snapshots at $t=2\times 10^4t_0$ confirm our predictions, with stationary dots/stripes at small $\alpha$ and traveling waves/dots at large $\alpha$. Simulations use $[100l_0,100l_0]$ periodic domains with homogeneous initial conditions plus small perturbations. Increasing $\text{Pe}_C$ shrinks the phase separation region; increasing $\alpha$ expands the oscillatory regime.
• Figure 4: Increasing chemical uptake rate ($\text{Da}$) and particle-to-chemical diffusivity ratio ($\alpha$) systematically alter the phase separation and oscillatory instability regions. (a-b) At small $\alpha = 1$, increasing $\text{Da}$ from $0.1$ to $0.5$ shrinks the stationary instability region, demonstrating that faster chemical uptake suppresses phase separation. (c-d) At large $\alpha = 10$, increasing $\text{Da}$ similarly contracts both the phase separation region and the oscillatory instability region (colored surface).
• Figure 5: Amplitude equation analysis quantitatively predicts the stability boundaries and bifurcation types of stationary patterns. (a) Phase diagram in $\text{Pe}_C-\mathrm{Da}$ space with $\text{Pe}_R=10^{-3}$, $\phi_0=0.775$, and $\alpha=2$, showing pattern morphologies from simulations of the weakly nonlinear model [Eq. \ref{['eqn::perturbation']}]. Stability boundaries predicted by the amplitude equation (solid curves) separate uniform (purple), stripes (yellow), and dots (green) solutions, with excellent agreement with simulations. Bistable regions exist where dots coexist with either stripes or uniform states. Below the red dashed line ($\sqrt{\mathrm{Da}\cdot\text{Pe}_C \cdot\tilde{c}_0}=\mathrm{Da}\cdot\phi_0$), patterns coarsen to system size. Simulations are performed on a $[200l_0,200l_0]$ periodic domain and solved on a $512\times 512$ grid. The snapshots are taken at $t=10^5t_0$. These parameters are selected such that no oscillatory patterns emerge. (b) Hysteresis at the uniform-to-dots transition [$\mathrm{Da}=0.6$, double-headed vertical arrow in (a)] demonstrates a saddle-node bifurcation. We calculate the amplitude from each simulation as $R_{\phi,\text{sim}} = \sqrt{\int{(\phi(\mathbf{x})-\phi_0)^2d\mathbf{x}}/(6\int{d\mathbf{x}})}$. Ramping $\text{Pe}_C$ down (blue) shows that the pattern persists below the stability threshold, while ramping up (orange) shows a delayed onset. Amplitude follows the fitted curve $R_\phi = R_0 + R_1 \sqrt{\text{Pe}_{\text{C,dots}}-\text{Pe}_C}$ (dashed line), where $R_0$ and $R_1$ are fit parameters; $R_0$ is nonzero at the bifurcation, indicating saddle-node behavior.