Spontaneous propulsion of an isotropic colloid in a phase-separating environment

The motion of active colloids is generally achieved through their anisotropy, as exemplified by Janus colloids. Recently, there was a growing interest in the propulsion of isotropic colloids, which requires some local symmetry breaking. Although several mechanisms for such propulsion were proposed, little is known about the role played by the interactions within the environment of the colloid, which can have a dramatic effect on its propulsion. Here, we propose a minimal model of an isotropic colloid in a bath of solute particles that interact with each other. These interactions lead to a spontaneous phase transition close to the colloid, to directed motion of the colloid over very long timescales and to significantly enhanced diffusion, in spite of the crowding induced by solute particles. We determine the range of parameters where this effect is observable in the model, and we propose an effective Langevin equation that accounts for it and allows one to determine the different contributions at stake in self-propulsion and enhanced diffusion.

Figure 1

Snapshot of the system: $N=500$ solute particles and a colloid $\mathrm{C}$ (orange) in a periodic square box. Enlargement of the colloid where the reaction $\mathrm{A}+\mathrm{C}\to \mathrm{B}+\mathrm{C}$ takes place in the reaction area of radius $r_{\text{cut}}$. A particles are in violet, and B are in green. A particles interact with purely repulsive interactions, whereas the interactions between B particles are attractive (Lennard-Jones). The interactions between the colloid and the solute particles are repulsive. The red circle represents the domain $\mathcal{P}$ within which the colloid may interact directly with solute particles.

Figure 2

Mean squared displacement of the colloid for fixed $r_{\text{cut}}=7.5$ and different values of $\varepsilon$ (left); and fixed $\varepsilon =3$ and different values of $r_{\text{cut}}$ (right). Solid lines are guides for the eye.

Figure 3

(a) Autocorrelation of the polarization vector $\mathit{p}$ and (c) cross correlations between $\mathit{p}$ and $\mathit{\xi }$ as function of time for different values of the parameters $\varepsilon$ and $r_{\text{cut}}$. The legend is the same for both plots and given on panel (c). In the absence of reaction, the autocorrelation of $\mathit{p}$ decreases very fast and oscillates around zero. Inset of panel (a): Enlargement of the short-time dynamics, represented in a log-log scale. (b) Mean squared displacement of the colloid as a function of time, and contributions defined in Eq. (3), for $\varepsilon =3$ and $r_{\text{cut}}=10.5$. The parameters for the analysis of the Brownian dynamics trajectories (b) are $K=0.0425{\tau }^{-1}$ and ${\tau }_0=0.3{\tau }_p=49.5\tau$.

Figure 4

Diffusion coefficient as a function of the parameters $\varepsilon$ and $r_{\text{cut}}$ (obtained from the MSD data shown in Fig. 2).

Figure 5

Snapshot of a system displaying a dense cluster of solute particles around the colloid in the stationary state ($\rho =0.3, \varepsilon =3$).

Figure 6

Mean squared displacement of the colloid as a function of time, and contributions in Eq. (3), for $\varepsilon =3$ and $r_{\text{cut}}=10.5$.

Figure 7

Autocorrelation of $\mathit{\xi }$ as a function of time for different values of the parameters $\varepsilon$ and $r_{\text{cut}}$.