Modeling complex motility patterns for autophoretic microswimmers

TL;DR

Isotropic autophoretic microswimmers rely on spontaneous symmetry breaking of a self-generated chemical field to propel in viscous environments. The authors develop a fixed-grid pseudospectral framework that solves the fully coupled advection–diffusion–Stokes equations, with slip and a regularized stresslet emerging self-consistently from the evolving chemical field, avoiding explicit moving boundaries. The method reproduces both steady propulsion at low $Pe$ and complex motility at higher $Pe$, including diffusive chemical trails, quadrupolar flow features, and chemotactic avoidance in pairwise interactions, with quantitative agreement to experiments on active droplets. This framework supports scalable simulations of many interacting swimmers and provides a robust tool for predicting chemo-hydrodynamic phenomena and emergent collective behavior in autophoretic systems. It has broad relevance for designing and understanding chemically driven micromachines and their interactions in complex environments.

Abstract

Symmetry breaking is essential for biological microswimmers to achieve locomotion in viscous environments. Such asymmetry in the swimming mechanism enables the generation of directional forces that overcome fluid resistance, leading to efficient motion and complex interactions. As synthetic analogues, autophoretic microswimmers including isotropic active colloids and active droplets exhibit spontaneous symmetry breaking of a chemical field, which generates interfacial flows and drives persistent self-propulsion. Modeling these systems is challenging because the chemical concentration and flow fields are strongly coupled through nonlinear advective transport of the chemical species. In this work, we propose a new numerical framework for modeling isotropic autophoretic microswimmers whose propulsion arises solely from self-generated chemical gradients, without any imposed geometric or chemical anisotropy. The framework employs a high-accuracy pseudospectral method to solve the fully coupled advection diffusion Stokes equations, without prescribing any slip velocity model.Slip velocities emerge self-consistently from instantaneous concentration gradients at the particle surface, driving propulsion and inducing flow disturbances through a stresslet representation of force and torque free swimmers. This approach naturally captures nonlinear advection, chemo-hydrodynamic feedback, and many-particle interactions within a unified framework. We demonstrate that the model reproduces complex emergent behaviors observed in experiments, including disordered swimming at higher fluid viscosities and chemotactically guided pairwise interactions. At each stage, numerical predictions are quantitatively compared with independent experiments on active droplets, validating the proposed framework as a robust tool for studying autophoretic microswimmers.

Figures (6)

• Figure 1: Schematic of the autophoretic microswimmer model. (a) At low Péclet number $(\mathrm{Pe})$, diffusion dominates, producing an isotropic solute field with uniform gradients throughout the domain. The Cartesian components of the concentration field shown in panels a(ii)–a(iii) remain perfectly symmetric, and the slip obtained from the discretised surface of the microswimmer in a(i) exhibits no tangential variation, resulting in zero propulsion. (b) At higher , advection becomes significant and the solute distribution develops weak anisotropy. Panels b(ii)–b(iii) show the resulting asymmetric concentration gradients, while b(i) illustrates the corresponding surface slip, which now has a finite mean component and drives motion opposite to the region of strongest slip. (c) A zoomed-out view highlights the swimmer’s displacement relative to its self-generated concentration field. (d) Radial solute distribution $\phi(r)$ (red curve) around a microswimmer, modeled using a hyperbolic-tangent profile with surface concentration $\lambda$, particle radius $a$ , and decay length $\delta$ that sets the distance over which the concentration relaxes to the background value. The inset schematic shows the microswimmer of radius $a$ and the surrounding decay layer (beige annulus of thickness $\delta$).
• Figure 2: Comparison between simulation and experimental results for microswimmer dynamics and hydrodynamic signature at low $\mathrm{Pe}$ number.First column (Simulation): Trajectory of the microswimmer (a) color-coded with the non-dimensional swimming speed $\bar{v}_{\mathrm{d}}$, and (b) colour-coded with the swimming orientation as given by the orientation angle $\psi$ of the swimming velocity vector (inset). (c) The flow field generated by the microswimmer represented using streamlines and contour plot for the local velocity magnitude $\bar{u}$. The results shown here are for simulations at $\mathrm{Pe} \approx 4$. Second column (Experiment): (d)-(f) Corresponding experimental observations for $\mathrm{Pe} = 4$. Temporal variations of the (g) mean-squared displacement (MSD) and (h) angular autocorrelation function $(C_{vv})$ for the swimming trajectory as obtained from the simulations (red line) and experiments (symbols). (i) The contour plot of the magnitude of the difference between simulation and experimental values of the local flow speed around the microswimmer swimming along identical direction.
• Figure 3: Comparison between simulation and experimental results for the spatio-temporal evolution of the chemical trail ejected by the autophoretic microswimmer at low $\mathrm{Pe} \approx 4$.Top row (Simulation): (a) A simulation snapshot showing the self-generated chemical trail left behind by the microswimmer during its self-propulsion. (b) Kymograph depicting the temporal evolution of the chemical concentration $(\bar{C})$ around the microswimmer's circumference, with angular position $\theta$ measured counterclockwise from the $+x$-axis (inset in a). The white line shows the swimming orientation in terms of $\theta$. (c) Temporal evolution of the chemical concentration along a fixed straight line $\mathrm{AA^\prime}$ (shown in a), drawn perpendicular to the instantaneous swimming orientation and passing through the microswimmer's center of mass. Inset: log--log plot showing a $\propto t^{-1/2}$ decay of the chemical concentration at a fixed point O on $\mathrm{AA^\prime}$ . Bottom row (Experiment): (d) Fluorescence micrograph showing the chemical (filled micelle) trail left behind by the active droplet. (e) Corresponding kymograph of fluorescence intensity $I(\theta)$ (filled micelle concentration) around the active droplet periphery during its self-propulsion. (f) Evolution of the filled micelle concentration along a fixed line $\mathrm{AA^\prime}$ as in (d). The inset also shows the $\propto t^{-1/2}$ decay of the filled micelle concentration.
• Figure 4: Emergent motility of the microswimmer with increasing viscosity of the swimming medium culminating in increasing Péclet number. (a)(I-IV) Simulated microswimmer trajectories for increasing $\mathrm{Pe} \in \{6, 35, 58, 254\}$ color-coded by the non-dimensional propulsion speed $\bar{v}_d$. (b)(i-iii) Experimental trajectories for increasing $\mathrm{Pe} \in \{4, 15, 50\}$ corresponding to increasing viscosity of the swimming medium (achieved by adding increasing weight percentage of glycerol to aqueous TTAB solution- 0%, 40%, and 60%). The white circles representing the active droplet are not to scale. Statistical analyses of numerical simulation results: (c) Variations of the average swimming speed$(<v_d>)$ and its standard deviation $(std_{v_d})$ with increasing $\mathrm{Pe}$. (d) Probability distributions of the non-dimensional, instantaneous propulsion speed $(\bar{v}_d)$ for different $\mathrm{Pe}$. (e) Variations of mean squared displacement (MSD), non-dimensionalized by $a^2$, with time interval $\Delta t$ for increasing $\mathrm{Pe}$. (f) Variations of the angular autocorrelation function ($C_{vv}$) with time interval $\Delta t$ for increasing $\mathrm{Pe}$. The inset shows the corresponding velocity autocorrelation function $(VACF)$ , highlighting a monotonic decay of velocity correlations with time interval $\Delta t$.
• Figure 5: Chemical and hydrodynamic signatures at relatively large $\mathrm{Pe}$. (a) Top row (Simulation): The microswimmer’s trajectory, color-coded by non-dimensionalized propulsion speed $\bar{v}_d$, for $\mathrm{Pe}=58$, and the corresponding kymograph for the chemical concentration around the microswimmer's periphery over a portion of the trajectory. We have marked three instants in the trajectory and the kymograph over a typical reorientation event at large $\mathrm{Pe}$-- (I) an instant during which the microswimmer undergoes ballistic swimming prior to a local reorientation in the swimming direction, (II) an instant when the microswimmer is in a state of transient arrest, and (III) an instant when the microswimmer resumes its swimming following a reorientation event. The flow field (streamlines) generated by the microswimmer along with the corresponding distribution of the chemical concentration at each of the three instants are also shown here. (b) Bottom row (Experiment): The experimentally obtained trajectory of an active droplet for a comparable $\mathrm{Pe}=50$, along with the corresponding filled micelle kymograph over a part of the trajectory. The white circle representing the active droplet is not to scale. Here we have also marked three similar instants (i, ii, and iii) over a reorientation event as in the numerically obtained trajectory. The flow fields of the active droplet at these three instants are also shown separately using streamlines and the contour plots for the local flow speed $(\bar{u})$.