Chemotactic motility-induced phase separation

TL;DR

This work introduces a continuum framework combining motility-induced phase separation (MIPS) with collective chemotaxis in active Brownian Particles, showing that chemotaxis can dramatically suppress or alter MIPS. Through linear stability analysis and numerical simulations, it identifies three key dimensionless groups (\alpha, Da, and Pe_C') that govern stability and instabilities, including finite- and unbounded-wavelength and oscillatory modes. The study maps these results onto the Pe_R-phi_0 plane, reveals rich coarsening dynamics, and demonstrates a spectrum of dynamic states from stationary patterns to traveling waves. By connecting theory with living and synthetic systems, it provides quantitative guidelines for tuning chemotaxis to control phase separation in active matter, with broad relevance to cells and chemotactic colloids.

Abstract

Collectives of actively-moving particles can spontaneously separate into dilute and dense phases -- a fascinating phenomenon known as motility-induced phase separation (MIPS). MIPS is well-studied for randomly-moving particles with no directional bias. However, many forms of active matter exhibit collective chemotaxis, directed motion along a chemical gradient that the constituent particles can generate themselves. Here, using theory and simulations, we demonstrate that collective chemotaxis strongly competes with MIPS -- in some cases, arresting or completely suppressing phase separation, or in other cases, generating fundamentally new dynamic instabilities. We establish quantitative principles describing this competition, thereby helping to reveal and clarify the rich physics underlying active matter systems that perform chemotaxis, ranging from cells to robots.

Figures (8)

• Figure 1: Chemotaxis suppresses MIPS. (a, c, e) Phase diagram, which is typically parameterized by $\phi_0$ and $\text{Pe}_\text{R}$, as predicted by linear stability analysis for different $\text{Da}_0$ and $\alpha_0$. The black curve shows the limit of stability without chemotaxis, below which we observe conventional MIPS. The colored solid and red dotted curves show Boundaries 1 and 2, which are defined in the main text; different colors indicate different values of $\text{Pe}_\text{C}$. Boundary 2 is below the horizontal axis in (a). The region above both Boundaries is stable, with ABPs in the homogeneous state, while the region below either Boundary is unstable. The different instability types---finite (F) or unbounded (U), stationary (S) or oscillatory (O)---are denoted by the shaded, unshaded, non-hashed, and hashed regions, respectively. Dash-dotted and dashed curves indicate the boundaries between F/U and S/O instabilities, respectively. The linear stability analysis predictions are corroborated by full numerical simulations (Movies S2-S4), snapshots of which are shown in (b, d, f), which focus on the grey boxed regions shown in (a, c, e).
• Figure 2: Chemotaxis arrests phase separation and generates dynamic instabilities. (a) Phase diagram parameterized instead by $\alpha_0$ and $\text{Pe}_\text{C}$, holding $\phi_0=0.8$, $\text{Pe}_\text{R}=10^{-3}$, and $\text{Da}_0=0.5$ fixed. Different instability types and the boundaries between them, as predicted by our linear stability analysis, are indicated using the same labels as in Fig. \ref{['fig::ABP']}. These predictions are again corroborated by numerical simulations (Movie S7), snapshots of which are shown. Arrows show the local velocity field $\mathbf{u}$, with the scale indicated by the characteristic velocity $u_0 \equiv M_0/\sqrt{\kappa} \sim U_0$; velocities for which $|\mathbf{u}|<0.005u_0$ are not shown. (b-c) Dispersion relations $\omega(q)$ corresponding to $\alpha_0=2$ and $\alpha_0=8$ in (a); solid (dashed) lines show the Real (Imaginary) components. Insets zoom in on long wavelengths. (d) Magnified views of the contours of $\phi=\phi_0$ and $\tilde{c}=\bar{\tilde{c}}$ (the spatial average of $\tilde{c}$) for the small regions indicated by the dashed rectangles in the snapshots of (a) at $\alpha_0=8$. Different colors in (b-d) show the different values of $\text{Pe}_\text{C}$ corresponding to the simulations shown in (a).
• Figure S1: A plot of $(\tilde{\mathcal{M}}-\tilde{\mathcal{D}})^2(\tilde{q}^2)$ and two lines that pass through the origin and are tangent to the curve at $\tilde{q}_l^2$ and $\tilde{q}_r^2$. $\text{Da}=0.05$, $\alpha=1.5$.
• Figure S2: Evolution of the characteristic domain sizes that correspond to the simulations at $\alpha_0=2$ and $8$ and increasing values of $\text{Pe}_\text{C}$ in Fig. 2(a) in the main text.
• Figure S3: (a-b) Snapshots of simulations and (c-d) evolution of the characteristic domain size $R(t)$ at $\text{Da}_0=0.5$, $\text{Pe}_\text{R}=10^{-3}$, $\phi_0=0.8$, $\alpha_0=2$ (a,c) and 10 (b,d), and increasing values of $\text{Pe}_\text{C}$.