Pairing, waltzing and scattering of chemotactic active colloids

TL;DR

This work analyzes non-reciprocal, diffusiophoretic two-body dynamics of chemotactic active colloids in a planar setting. By solving the diffusing product field via a method of reflections and deriving slip-velocity–based equations of motion, the authors reveal rich behavior governed by the separation $R$, relative orientations $\Delta_1,\Delta_2$, and the propulsion-to-interaction ratio $h_0= a^{(1)}/a^{(2)}$, including bound states (active dimers), phase-locked binary swimmers, and scattering. The study systematically characterizes dynamics for a swimmer near a fixed source and for two mobile swimmers, mapping outcomes to state diagrams across swimmer designs and highlighting how isotropic versus anisotropic sources, as well as fluctuations, influence stability and transitions. The findings illuminate how long-range chemotactic attraction can balance phoretic repulsion to create stable configurations, or induce persistent orbits and chasing dynamics, with direct implications for experimental realization in microfluidic confinements and extensions to incorporate hydrodynamics or three-dimensional motion.

Abstract

An interacting pair of chemotactic (anti-chemotactic) active colloids, that can rotate their axes of self-propulsion to align {parallel (anti-parallel)} to a chemical gradient, shows dynamical behaviour that varies from bound states to scattering. The underlying two-body interactions are purely dynamical, non-central, non-reciprocal, and controlled by changing the catalytic activity and phoretic mobility. Mutually chemotactic colloids trap each other in a final state of fixed separation; the resulting `active dimer' translates. A second type of bound state is observed where the polar axes undergo periodic cycles leading to phase-synchronised circular motion around a common point. These bound states are formed depending on initial conditions and can unbind on increasing the speed of self propulsion. Mutually anti-chemotactic swimmers always scatter apart. We also classify the fixed points underlying the bound states, and the bifurcations leading to transitions from one type of bound state to another, for the case of a single swimmer in the presence of a localised source of solute.

Figures (13)

• Figure 1: Schematics summarizing the final states. Sign of phoretic angular velocity that rotates the polar axis varies with the distance $R$ between swimmers. When it is positive (negative) at all $R$ for both swimmers, they are mutually chemotactic (anti-chemotactic) colloids. Self propulsion drives chemotactic (anti-chemotactic) swimmer towards (away from) one another. A second contribution to centre of mass motion is from phoretic response to the chemical gradient. (a) Alignment between a chemotactic pair leads to formation of a dimer by a cancellation of relative velocity. (b) Unsuccessful swimmers run away from one another. (c) Swimmers revolve around a common centre while maintaining a fixed angular separation between their axes. This happens when the sign of the phoretic angular velocity changes at small $R$. (d) Anti-chemotactic swimmers always scatter off one another. (e) A mixed pair shows explicit signatures of non-reciprocal interactions where one swimmer chases the other to form a bound pair.
• Figure 2: 2D state diagrams categorizing steady state dynamics are constructed by varying paired combinations of polar mobility $\mu_{1}^{(i)}$, apolar mobility $\mu_{2}^{(i)}$ and total surface activity $a^{(i)}$, keeping other parameters and initial conditions fixed (for details, see Appendix A). Figs. (a)-(c) on the left show the trapped, orbiting and scattered states observed when a swimmer interacts with a source for three values of $h_0 = a^{(1)}/a^{(2)}$. In figs. (a) and (c) the source produces an isotropic districution of product while in fig. (b) it is anisotropic. Increasing $h_0$, equivalent to increasing the speed of self-propulsion, tends to eliminate bound states by causing scattering as can be seen by comparing figs. (a) and (c) which differ in $h_0$ only. Fig. (d)-(f) on the right show bound states for two mobile swimmers showing active dimers, binary-swimmers and scattering.
• Figure 3: Schematic showing the relative position of the two swimmers. Swimmer velocities point in directions different from the line joining their centres, leading to non-central and non-reciprocal interactions. By symmetry, equations of motion for colloid position and polarity can depend only on the relative orientations $\Delta_{1,2}$ and separation $R$. Fixed points of this two-body system is determined by stationary values of the three relative parameters while actual motion occurs in the six dimensional space leading to interesting oscillatory states.
• Figure 4: Typical paths traced by a swimmer near isotropic and anisotropic sources are shown in figs. (a) and (b) respectively. The coloured markers represent swimmer position, arrows represent polar axis. Marker colour changes from blue to red to with time. An isotropic source produces a radially symmetric effective potential plotted in fig. (c) as a function of $R$ for different choices of $h_0$. The swimmer is strongly confined for larger self propulsion. An anisotropic source traps the swimmer preferentially along its axis of symmetry producing an anisotropic potential plotted in fig. (d) for motion along $\beta =0,\pi$. The mobility and catalytic coat designs are $\{ 0.3,-0.5, 0.4 \}$ and $\{1, -1, 0\}$ respectively. The catalytic coat of source is $\{1,1,0\}$. $h_0 = 0.03$ for fig. (a), (b) and (d).
• Figure 5: Swimmer orbits around the source tracing periodic circular orbits in fig. (a) which become non circular as the source is made an-isotropic in fig. (b). The parameters for the source are same as in Fig. \ref{['Isotropic']} and the parameters for $\{ \mu^{(1)}_{\ell}\}$ of the swimmer are $\{0.3,-0.5,1\}$. In the case of an anisotropic source, the fixed point exists in three dimensional space spanned by $\Delta_{1},\beta,R$. In (c) we show two dimensional projections in $\Delta_1-R$ plane for $h_0=0.04$ (blue line)and $h_0=0.06$ (gray line). The red point is the fixed point for an isotropic source for $h_0=0.04$.